As I understand it, the name means "drunkard." It sounds very much like the Spanish word for drunk "borracho," so I tend to believe this explanation. The word is also supposed to refer to a type of skin for holding wine.
In addition, there is at least one line in the play where Borachio himself says something that would support this interpretation of the meaning of his name. This is in Act III, Scene 3 (around line 95) where he says
I will, like a true drunkard,
utter all to thee.
I believe that, by saying this, he is making a pun on his name.