According to many scholars it appears that it does.
The Online Etymology Dictionary offers this origin:
1551, from M.L. from Arabic al jebr "reunion of broken parts" as in computation, used 9c. by Baghdad mathematician Abu Ja'far Muhammad ibn Musa al-Khwarizmi as the title of his famous treatise on equations ("Kitab al-Jabr w'al-Muqabala" "Rules of Reintegration and Reduction"), which also introduced Arabic numerals to the West. The accent shifted 17c. from second syllable to first. The word was used in Eng. 15c.-16c. to mean "bone-setting," probably from the Arabs in Spain.So what Hebrew root could be related to "bone-setting"?
There are those try to connect it to the root חבר meaning "to connect", but as Mike Gerver points out in this very important Mail-Jewish post:
In fact, Arabic j corresponds to Hebrew gimmel, not to chet. (The Arabic letter is called "jim," and pronounced as a hard g in some dialects of Arabic, while Jews from Yemen pronounce gimmel like j, I think only when it has a dagesh.) According to my Arabic dictionary, jabara means not just "join together," but "force together," and is used for setting broken bones. So Hebrew gimmel-beit-resh would seem to be a much more likely relative.
And indeed, the American Heritage Dictionary connects jabara to the root גבר :
To be(come) strong, prevail, work. West Semitic variant (assimilated) form gbr.However, Dr. Solomon Gandz of Yeshiva University wrote in this 1926 article that while algebra is connected to גבר, it may not be via "bone-setting":
1. Gabriel, from Hebrew gabrîl, my strong one (is) God, from gabrî, my strong one, from gabr-, presuffixal form of geber, strong one, man, from gabar, to be strong
2. algebra, from Arabic al-jabr, the might, force, restoration, from jabara, to force, restore, set (bones).
There are still remnants in the mathematical literature suggesting that in olden times the term al-jabr alone was used for the science of equations, and the term al-jabriyyun was taken for the masters of algebra. On the other hand the term al-muqabalah alone, according to its real meaning of "putting face to face, confronting, equation," seems to be the most appropriate name for equations in general. With these difficulties in mind, the writer undertook to search out the real meaning of jabara in the related Semitic languages. Now the Assyrian name gabru-maharu means to be equal, to correspond, to confront, or to put two things face to face; see Delitzsch, Assyrisches Handworterbuch, under gabru and maharu, pp. 193, 401, and Muss-Arnolt, Assyrian Dictionary, under gabru and maxaru, pp. 210, 525. From the first of these we have the etymology of the Hebrew geber and gibbor. Geber is the mature man leaving the state of boyhood and being equal in rank and value to the other men of the assembly or army. Gibbor is the hero who is strong enough to fight and overcome his equals and rivals in the hostile army. Gabara =jabara, in its original Assyrian meaning, is therefore the wrresponding name for the Arabic qabala (verbal noun muqabalah), and an appropriate name for equations in general.
Whether or not jabara meant "bone-setting" or actually referred to equations, many today would view someone who can master algebra as a real gibor גיבור - hero...