I love that you’re going for something unique, but Bungee doesn’t feel very usable as a baby name to me. Both spellings are likely to be read like the word “bungee” (as in bungee cord), so it could be hard for people to take seriously in everyday life, and he’d probably get a lot of jokes/comments. And honestly, I weirdly prefer the bungee cord pronunciation (if it weren’t for the association), as bun-ghee makes me stumble a bit when I say it out loud. Also “-ee” ending doesn’t read feminine to me at all since many boys names have ee endings.
[name_f][/name_f]I agree with @Madsummer[name_f][/name_f] that [name_m]Benji[/name_m] is a great alternative. Some more ideas you might like:
[name_f][/name_f][name_m]Reggie[/name_m], [name_m]Quincy[/name_m], [name_m]Gideon[/name_m], [name_m]Bastian[/name_m], [name_m]Bowie[/name_m], [name_m]Hugo[/name_m], [name_m]Magnus[/name_m], [name_m]Angus[/name_m], [name_m]Monty[/name_m], [name_m]Benson[/name_m], [name_m]Briggs[/name_m], [name_m]Bingham[/name_m], [name_m]Burgess[/name_m], Barnaby/Barney, [name_m]Eugene[/name_m], [name_m]Beau[/name_m] (and [name_m]Bogdan[/name_m] if you want something very bold and old-world!)
I also thought of Bounty, Bundle, and Bugle (“BYOO-gull”), but those have such strong word associations that they might be even trickier than Bungee.
[name_f][/name_f][name_f]Hope[/name_f] this helps!