The name Abel in the Bible
There are two different Hebrew names in the Bible, which transliterated into English both came out as Abel. We'll treat them separately and call them Abel I (הבל, probably pronounced as Habel) and Abel II (אבל, probably pronounced as Abel). The difference between the two Abels is that Abel I is spelled with the letter ה (he), while Abel II is spelled with the letter א (aleph). These two letters are notorious for alternating, but in the case of the two Abels, both names are derived from common Hebrew words, not one word with two different spellings:
The name Abel I in the Bible
There's only one person named Abel (הבל) in the Bible. He is the second son of Adam and Eve; brother of Cain and Seth (Genesis 4). He holds the dubious honor of being the first person to die, to die due to a violent act, to die as the result of a crime, and to die childless. His brother and murderer Cain has seven generations of offspring, but his whole lineage dies just like his brother Abel, in the flood of Noah (Genesis 6).
Etymology of the name Abel I
The name Abel comes from the Hebrew verb הבל (habal), meaning to act emptily or become vain:
The names of important characters in Genesis are usually explained while they are given, but not Abel. Here is an individual who is named Whiff, who is not even worthy of an name-explanation, and who has himself done in by his brother Cain, before he can even say a word! If God hadn't accepted his offering (Genesis 4:4) and Jesus hadn't proclaimed him righteous (Matthew 23:35, spelled Αβελ) Abel would make for a very sad character. Which at once raises the question of what he is doing there so early in the narrative, as well as the meaning of this entire nonpareil scene.
Abel I meaning
Jones' Dictionary of Old Testament Proper Names reads Vanity. NOBSE Study Bible Name List reads Breath. BDB Theological Dictionary, strict as ever, reserves for the name Abel a whole separate root but can't give a translation.
Perhaps, says BDB, Abel has to do with the Assyrian word ablu, meaning son. J. Wellhausen too derives of the root treated above but BDB deems this "not probable."
It is, however, most probably that no one in a classical Hebrew audience would be aware of BDB's objections or thought that Abel might mean anything else than Breath or Vanity, also because Abel's mother, father and two brothers have names that are recognizable common Hebrew words.
That is, of course, unless in the context of the story, the name הבל (Abel I) was understood to be just an alternate spelling of the name אבל (Abel II), and meant Mourning (see below). Since the name of Cain has to do with acquisition, the Abel may have to do with sorrow, and the story of Adam and Eve is overly obvious more allegorical than historical, we can almost hear the words of the Preacher, who not only said "Vanity of vanities, all is vanity" (Ecclesiastes 1:2), but also, "increasing knowledge [Cain?] results in increasing pain [Abel?]" (Ecclesiastes 1:18).
The name Abel II in the Bible
The name אבל is assigned only to locations; most of them with names that are constructions of which Abel is the first part. But there is one town that's called just Abel, or so it seems. The town called Abel occurs first in 1 Samuel 6:18, although most scholars agree that this is probably a text error and should read אבן, ('eben), meaning stone. Then there is a town named Abel in 2 Samuel 20:14 and verse 18, but in verse 15 the text speaks of Abel-beth-maacah. So either Abel and Abel-beth-maacah are adjacent and similarly named towns (like, say, Beverly and Beverly Farms in Massachusetts) or they are the same town known by a long, complete and a short, truncated name (like, say, Philadelphia a.k.a. Philly). We really don't know.
Etymology of the name Abel II
The name Abel II comes from the root group אבל:
Abel II meaning
For a meaning of the name and name-element Abel II, BDB Theological Dictionary consistently reads Meadow. Jones' Dictionary of Old Testament Proper Names reads Mourning for Abel, but varies the interpretation of the compound names. NOBSE Study Bible Name List consistently reads Meadow. But if HAW Theological Wordbook of the Old Testament is correct, the name Abel means Stream or Brook.