There are a number of theories.
One theory, popularized by Yosefon (or Josippon), and which is widely viewed as a folk-etymology is that the word מכבי makabi is an acronym of the verse (Shmot 15:11)
מִי-כָמֹכָה בָּאֵלִם יְ-ה-וָ-ה
"Who is like you among the mighty, God"
A less well known acronym, as mentioned here, is:
מתתיהו כהן בן יוחנן
"Matityahu Kohen son of Yochanan"
This site, which discusses the origin of the term, mentions the theory that the name could come from מכבני :
The scholar and poet Aaron Kaminka (1866 - 1950) thinks the name is a corruption of Machbanai, a leading commando in the army of King David (I Chron 12:13). David had 12 commandos from the tribe of Gad, who "separated themselves to David to the stronghold in the wilderness, mighty men of valour, men trained for war, that could handle shield and spear, whose faces were like the faces of lions, and they were as swift as the roes upon the mountains" (I Chron 12:8). David has always been a role model for Jews, and it may be that Judah's father, Mattathias, saw in his son the embodiment of an ancient Davidic hero.
Klein writes that perhaps the correct spelling should be מקבי - which derives from makevet מקבת - "hammer". Klein defines it as:
the name having been given to Judah the Hasmonean, because he struck the Syrians as with a hammer. Compare the name of Charles Martel (689? - 741) ruler of the Franks, from French martel (=hammer) who was so called because he struck the Moslems as with a hammer.An additional theory, also attributed to Klein here is that:
1375, from L.L. Maccabæus, surname given to Judas, third son of Mattathias the Hasmonean, leader of the religious revolt against Antiochus IV, 175-166 B.C.E., usually connected with Heb. maqqabh "hammer," but Klein thinks it an inexact transliteration of Heb. matzbi "general, commander of an army."
We can connect מצביא and מקבי either through Hebrew/ Aramaic, where tzade and kuf interchange or through Greek / Latin, where the letter C can be either be hard (like K) or soft (like S).
Additional theories can be found in this Jewish Encyclopedia article.
One interesting English word that derives from makabi is macabre:
c.1430, from O.Fr. (danse) Macabré "(dance) of Death" (1376), probably a translation of M.L. (Chorea) Machabæorum, lit. "dance of the Maccabees" (leaders of the Jewish revolt against Syro-Hellenes, see Maccabees). The association with the dance of death seems to be via vivid descriptions of the martyrdom of the Maccabees in the Apocryphal books. The abstracted sense of "gruesome" is first attested 1842 in Fr., 1889 in Eng.
(More can be found about the etymology of macabre here.)
But in Modern Hebrew the word has a much more upbeat connotation. Maccabi is the name of a large sports organization, and Rosenthal offers these words in his slang dictionary:
מכביזם - makabism - the atmosphere attributed to the fans of Maccabi Tel Aviv's soccer and basketball teams, characterized by competitiveness
מכביסט - makabist - a fan of Maccabi Tel Aviv's soccer and basketball teams