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: