A few of its words are about a girl/woman, but it would be easy enough to change them to suit a boy/man...
There's Sunrise Sunset... although the 2nd verse is about a wedding, so not quite so appropriate; maybe just the first verse? Or maybe adapt the 2nd verse a bit to fit a bar mitzvah? Perhaps something like:
He looks so confident and handsome All that we might have dreamed, and more Now we can see his own dreams growing, strong and sure
May he have much to give the future May he know happiness and love May he be blessèd by the Lord above
Sabbath Prayer (also from Fiddler on the Roof) also seems very apt for the occasion, although it's not about growing up. (It also would need *slight* adjusting to make it appropriate for a young man.)