I will keep the question simple. Who is the little horn that 'has eyes' and is 'speaking' blasphemous things in Daniel 7?
Short answer: Antiochus Epiphanes.
An answer for the 'little horn' rests on which presuppositions the reader is willing to make about the book of Daniel. Having at least one presupposition is inevitable, no matter who you are, and that affects the way one reads the entire book. My presupposition is this: Daniel is about kingdoms, and the author names all of the kingdoms that figure into his visions.
The first six chapters, the stories about Daniel and his fellow Jews during their activities in the royal courts of foreign kingdoms (aka the court tales), establish two themes in union together. First, God controls the rise and fall of kingdoms. (5.21) Second, it is better to serve this God than submit to pressure or pain to commit idolatry or other sins. (3.17-18)
By the time these two themes are established after six stories that illustrate them, the narrative begins to elaborate on the dream from chapter 2. This elaboration comes in the form of Daniel's dreams and visions, and an angel interpreting them to Daniel. What we read in chapters 2, 7, 8, 9, and 10-12 are all ultimately parallel and complementary descriptions of the same thing: the rise and fall of a series of kingdoms, the last of which is oppressive toward 'the saints', before that kingdom is in turn overthrown.
Again, my presupposition is that the author names all of the kingdoms in the series he describes: the first is Babylon (2.36-38), the second is a unified Media and Persia (8.20; 10.20a; 11.2a), and the third is Greece led by Alexander (8.21; 10.20b; 11.2b). To any historian of the ancient near east, what the author is portraying is very clearly the series of kingdoms that held power over the Jewish people: Babylon (605-538 BC ), then Media-Persia (538-330 BC), then Alexander's Greece (330-323 BC).
This is where most presuppositions diverge, so that the fourth kingdom is thought to be the Roman empire, or the Roman papacy, or Islam, or what-have-you.
The details in Daniel 8 and 11, however, keep the historical context in the line of kings that followed Alexander. Both chapters mention how the king of Greece will suddenly die (as Alexander did), with his empire splitting four ways (an overly simplistic summary, but true enough). This is the fourth kingdom (323-140 BC). Chapter 11 describes the feuding of these successor kings, the Diadochi, in great detail, events that occurred in the fourth, third, and second centuries BC; primarily the Ptolemies (south of Israel, in Egypt) and the Seleucids (north of Israel, in Syria).
This brings us to the main question. The culminating events of each section are complementary details of the same historical events. Each section even uses several of the same words or phrases: 'little horn' of the fourth kingdom in chapter 7 and 8, the 'desolation' caused by transgression/abomination, and halted offerings in chapters 8 and 9 and 10-12, etc.
The descriptions of historical events in chapter 11 are too precise to be talking about anyone other than Antiochus Epiphanes and the events of 170-164 BC.  The overt parallelism with the dreams and visions makes it certain that he is the 'little horn'.
Critical scholars opine that Daniel was written during or shortly after Antiochus' reign. The two themes from the first six chapters illustrate the situation of the author and his fellow Jews: they are under the rule of an evil kingdom and must not give in to pressure or pain to commit idolatry or other sins, as they saw Antiochus doing to them. Further, it seems this was the 'original' interpretation of Daniel just a few decades after it was written (see note 2 below). Whether it became a widely recognized one is another issue, though. As early as the first century, some Jews (like Josephus) thought Daniel was a prophecy about Rome.
 The critical history doesn't have Babylon take the Jews into exile until 597 BC. However, the first chapter of Daniel dates this to 605 BC, so I'm following the author's understanding of history, since it's his book.
 The first several chapters of 1 Maccabees, written decades after the Maccabean revolt, make a few direct references to Daniel's descriptions, but does so without saying so. These include references to the 'abomination of desolation' carried out at Antiochus' order, the account of Antiochus' war against Egypt with ships and mounted troops, and a somewhat dissimilar description of three of Israel's neighboring countries siding against them.
In order to identify the symbolic in Daniel it is interesting to take note of their reception and similarities within the Apocalypse of John along with the fact that according to Matthew (ch.24) Jesus identified events in 1st century Jerusalem with parts of Daniel's prophecy.
This last of the four animals representing political powers (kingdoms) is the most devastating, the Roman, that killed God's anointed and certainly ten thousands, perhaps even hundred thousands of Jews, and destroyed holy city and temple.
The ten horns standing for successive kingdoms could mean those powers that came up after the Roman Empire fell apart. The Kaisers (gr. kaesaros for lat. cæsar) e.g. led the so called Holy Roman Empire up to the 19th century. The title (such as Czar in Russia) was used by different powers for their kings until the 20th century. Roman law formed the basis for the law of most western states up to our days, even the Roman catholic church law is in many of its (non-biblical) aspects derived from there. Republican state constitutions took name and principles from the res publica of ancient Rome. Fascism also borrowed symbols and ideas - its martialism and hybris - from there.
If the ten horns stand for these states after Rome, what is that other different small horn that is speaking arrogantly against God and harassing the holy ones standing for?
When looking into Revelation the ten horns are there. The beastly animal here has traits of all other animals, though, and seven heads, here even more representing a totality of political powers. Where is the different horn? No additional horn but another animal is introduced here (ch.13). With horns like those of a male lamb but a mouth like a dragon's mouth speaking things like the small and different horn did in Daniel's vision.
What things? Here (Revelation 13) forcing the worship of images / idols of political power, there (Daniel 7) changing times and law.
What kind of power is that, different from the others yet having great influence through speech and manipulation?
Towards the end of the visions of Revelation (ch.17) again a wild animal with ten horns appears, here explicitly representing the kings of the earth. The small horn? Not there, but again something else instead: the infamous whore, mother of whores, being a great city by which she is ruling and deceiving and whoring with the powers of the earth.
Very likely it is about religion, cultic idolatry affiliated with state power, in its different manifestations:
The United States.
To explain this, we first must understand that the Context is Figurative/representitive of another reality, as the 'natural'(historical/grammatical) method applied to this text makes no sense and "does violence to the text".
We also must employ an important rule of Figurative Interpretation:
If the symbol is interpreted in one place in Scripture, then the SAME INTERPRETATION must be applied throughout Scripture(both Old and New Testaments), unless God changes it(gives the same symbol a different meaning).
In Dan. 7:7, Daniel sees,"A fourth Beast, dreadful and terrible, and strong exceedingly; and it had great *IRON(*not gold, not silver, not brass) TEETH.
To understand IRON, we turn to Dan. 2:40,"And the FOURTH KINGDOM shall be strong as IRON"-this conincides with Nebuchadnezzar's Statue (vs 33*)"...had legs of IRON*".
Since Figurative "Iron"="Iron", then "Fourth Kingdom"="Fourth beast", and since these kingdoms and beasts are Progressive(one succeeding another), then the 4th Beast HAS TO BE ROME! The Greeks were the 3rd Beast in succession, represented by BRASS, the 2nd Beast was the Mede/Persians(Silver), and Babylon was GOLD-which Daniel indicated was Babylon in Dan. 2:38.(Thou art this head of GOLD.)
The TEN HORNS which came from Rome are the very same TEN HORNS we see in Rev. 12:3, 13:1, and 17:3. The TEN HORNS are interpreted for us in Rev. 17:12,"The ten horns are 10 kings, which have received NO KINGDOM AS YET(during the time of John); but receive power as kings one hour with the beast." Since the Context is Figurative, the 'one hour' doesn't mean 60 minutes; but it is a relatively short period of time, compared to the rest of the Heads of which the Horns rest on.
These Horns can also be interpreted 'kingdoms', since in Rev. 12:3 they appear "without crowns"; in Rev. 13:1, they appear "with crowns"-signifying their arrival to power.
But back to Dan. 7:8, and 25-26, which are the explanation of it*:"...another shall rise after them and subdue 3 Kings*" Notice that it didn't say 'entirely remove them', and we see in Revelations, it is the Beast with 7 Heads and TEN HORNS, not '8 horns'.
Therefore, 3 HORNS from the HEAD of ROME were 'subdued' to make room for this 'little horn'. FRANCE, SPAIN, ENGLAND, all former colonies of Rome; now nations in their own right, WERE MOVED(subdued) to make room for this "little horn". The 'little' means 'brief', insignificant(for a season), lesser(in terms of gathering colonies and expanding their rule globally). Compared to the French, British and Spanish Empires, the US is insignificant.
But it is DIVERSE(different) from the others: The US ORIGINATED AS A DEMOCRACY, which none of the other "horns" have done; consequently, it's government is expressed as "The Will Of The People", vs Divine Decree which the other "Horns" originated as. THERE IS NO ONE NATIONALITY GROUP OR RACE, IT IS THE MIXTURE OF ALL RACES AND NATIONALITIES. It is as if to say the "Tower of Babel" being built all over again. There was NO NATURAL GEOLOGICAL BORDERS which 'separate' us; the US occupied a land that was largely barren of people.
The GREAT BOASTS are what distinquish this Horn:,"We will conquer space", "We will conquer disease, poverty, drug abuse, over-population, global warming," etc. WE will adjudicate and resolve all conflicts, including Israel's conflict with it's Arab neighbors and Palestinians. We have exported our goods and entertainment until they become the standard by which all others are measured 'corrupting the earth' as we have progressively lowered the bar of acceptable standards.
The "Names of Blasphemy" are 'Man is god': this is what Jesus was put to death for, Matt. 26:65 spells this out,"...He has spoken blasphemy; what further need we have we of witnesses? Behold, now ye have heard blasphemy". This is after Jesus said,"Hereafter shall ye see the Son of Man sitting on the right hand of power, and coming in clouds of heaven."
Secular Humanism is the religion of "Man as god." This has become increasingly the 'religion' of this land-"...Having a 'form' of godliness, but denying the power thereof."(2 Tim. 3:5) We see this in the laws being changed to permit those things God calls evil, and exclude those things God calls good. This is how the "saints are worn out"(Dan. 7:24)
The "spirits of man" identify the 'eyes' of this horn, versus the "Spirits of God" which are the "7 eyes, which are the 7 Spirits of God sent forth into all the earth."(Rev. 5:6). The eyes- "see"(know/understand). Just as God's eyes "see" all and understand all there is, so does this little horn "attempt" to see all, and understand all there is-from man's perspective. It is no accident that the major reasearch, and greatest minds from around the world are brought here, to 'solve' mankind's problems.
Conclusively, the 'little horn', in this instance represents the United States.
Since "little horn" has been used in regards to Antiochus Epiphanes, and I have stated that this Figurative symbol MUST be interpreted the same, unless it is interpreted differently in the text, I will explain how the "little horn" CAN mean Antiochus Epiphanes in Dan. 8:9(NOT IN DAN. 7:8) and also refer to the United States.
To understand this, one must understand the principle of Dual Fulfillment: just as the prophecies of Christ include His 1st time appearance as the Suffering Servant of Isa. 53, and yet also refer to His Coming as King and Lord, reigning in Jerusalem(Zech. 14:3-4), so also the "little horn" must be understood.
The "Little Horn" of Dan. 8:9 comes from one of the 4 Notable Horns; which ARE NOT THE SAME AS 10 HORNS, rather, they come from the 4 Generals(Seleucus, Ptolemy, Cassander, and Lysimachus), that ruled in Alexander the Great's stead, in the dividing of his kingdom. Their kingdoms were North, South, East, and West, and Seleucus was "south and east-towards the Pleasant Land".(vs 9)
Antiochus Epiphanes was indeed a king who 'connived' his way to power(Dan. 11:21), was AGAINST the Holy Covenant(vs 28-he put a statue of Zeus in the wing of the Temple, causing the "Abomination that makes Desolate", commanded all the people to renounce God and worship his pantheon-at the pain of death(this is found in 1 Maccabees). He of course died, after Judas Maccabees and his brothers repeatedly defeated him, casting him out of Judea, where it is reported he went to his death being eaten alive by worms(2 Maccabees)
But this DOES NOT explain the "king" of the time of the end(Dan. 11:36-45), after which in Dan. 12:1, there comes a 'Time of Jacob's Trouble' and everyone that is "written in the book" shall be delivered. This is also referred to when Jesus said,"When you shall see the Abomination of Desolation spoken of by Daniel the prophet, stand in the holy place*(whoso readeth, let him understand*)(Matt. 24:15)
It CANNOT be Antiochus Epiphanes-he died 160+ years prior to Jesus's statement. So it must be someone else, and the key is Dan. 11:37,"..he shall magnify himself above all." This is Secular Humanism-it exalts itself "Mankind" over any other force-it doesn't recognize any other. Since this is the "boast" of the little horn" of Dan. 7:8, it is conclusive that the United States is not 'satisfied' with adjudicating the rest of the world's problems, it also wants to "solve" the problems of Israel and their neighbors-by putting in a plan which will divide Jerusalem(read-Road Map for Peace) and leave Jerusalem in the hands of the UN.
This of course will put Israel at the mercy of her enemies, which have sworn to annihilate her. It is then that all the prophecies will be fulfilled, and the Great Battle will take place-of which the Lord will descend with His Armies(Joel 3) and defeat them.
It is amazing in my lifetime how far down the road we have come to fulfilling this prophecy.