In The Etymologies in volume 5 of The History of Middle-earth, the name Bór is interpreted as Elvish, meaning 'steadfast, trusty man, faithful vassal'. This fits well with the fact that Bór and his sons remained faithful to the Elf-lords they served, and the natural assumption would therefore be that Bór gained an Elvish name meaning 'faithful' due to this service.
This neat interpretation is complicated somewhat by comments elsewhere in the same volume (in The Lhammas) where we're told that the folk of '...Bor, and of Uldor ... were different in speech, but that speech is lost without record other than the names of these men.' According to this source, then, Bór was a name from a lost Mannish language, without a specified meaning.
These two perspectives are not easy to reconcile, and presumably simply reflect a change of mind by Tolkien, though it is difficult to be sure which explanation takes precedence. (At a stretch, we might imagine that this Easterling was named Bor in his original native language, and that this was adapted by the Elves to the conveniently close Bór, 'faithful', though it is nowhere implied that this was Tolkien's intention.)