The Greek word behind remorse/repent is μεταμεληθεὶς, pronounced metameletheis, coming from metamelomai. It is found six times in the New Testament: Matthew 21:29, 32; 27:3; 2 Corinthians 7:8 (twice); and Hebrews 7:21 (quoting from Psalm 110:4 where it translates the Hebrew nacham). It is uniformly translated as "repent" in the KJV.
While some may say that metameletheis means a mental regret without going deeper, that can't be seen from the uses in the New Testament.
He answered and said, I will not: but afterward he repented, and went. (Matthew 21:29)
This verse shows more than just mental regret. It shows a change in action.
For John came unto you in the way of righteousness, and ye believed him not: but the publicans and the harlots believed him: and ye, when ye had seen it, repented not afterward, that ye might believe him. (Matthew 21:32)
This repentance is a prerequisite to believing the preaching of John the Baptist.
Then Judas, which had betrayed him, when he saw that he was condemned, repented himself, and brought again the thirty pieces of silver to the chief priests and elders, (Matthew 27:3)
Though we are examining this verse, it should be noted that by placing "himself" after it, the KJV translators might be implying that Judas' repentance was mental regret. That is, Judas did not repent to God, he repented to himself. However, that is not seen from merely the form of the Greek word.
Literary English allows for "repented" and "repented himself" to mean the same thing. Use of one or the other is merely a stylistic choice. However, this is the only time in the NT that the KJV placed a reflexive pronoun after "repented." Between the two Greek words for "repent" in the NT, there are about 40 occurrences. While it could simply be a stylistic choice, this being the only place to change styles sticks out as odd.
Matthew Henry's Commentary says of this verse, "He repented himself; that is, he was filled with grief, anguish, and indignation, at himself, when reflecting upon what he had done." He then goes on to compare how Peter's repentance led to salvation while Judas' to destruction. With Henry's commentary being ~100 years after KJV, his understanding of the phrase would be similar to theirs.
For though I made you sorry with a letter, I do not repent, though I did repent: for I perceive that the same epistle hath made you sorry, though it were but for a season. (2 Corinthians 7:8)
Paul seems to use both meanings of the word in this verse. He does not repent to God (the first) but does regret that his actions brought sorrow. Perhaps his regret is that the sorrow lasted only a short time.
(For those priests were made without an oath; but this with an oath by him that said unto him, The Lord sware and will not repent, Thou art a priest for ever after the order of Melchisedec:) Hebrews 7:21
The Lord will never change His mind, though it is also impossible for God to repent of sins. (The Hebrew nacham likewise ranges in meaning from regret to repentance.)
Metameletheis may mean either "repentance to God" or "regret to oneself." Had the synonym metanoeo been used in Matthew 27:3, that would more certainly indicates moral action. Thayer's Lexicon says the higher nobility of metanoeo is seen in how metanoeo is often in the imperative (while metamelomai never is). Metanoeo appears 34 times in the New Testament (and is always "repent" in the KJV). Metanoeo is often used in ways that indicate it is a change of heart and mind. This will either be stated ("repent... and pray to the Lord that he may forgive you," Acts 8:22) or seen in the person's actions of contrition ("repented long ago in sack cloth and ashes" Matthew 11:21, paralleled in Luke 10:13; "bring fruits worthy of repentance" Luke 3:8).
In conclusion, the word used in Matthew 27:3 for Judas' action is used in the New Testament for both mental regret (2 Corinthians 7:8) and active repentance (Matthew 21:29, 32; 2 Corinthians 7:8), with Hebrews 7:21 able to mean either.