Boy is a Noun, not an acronym, so it does not have any full form. A man or a male could be called a boy id he is not married and is quite young.
Boy does not have any full form, but there could be some funny full forms of it like the one written below:-
B – Badmashiyon mein sab se aage
O – Ollu Ki Tarah raat mein Jaage
Y – Yaariyan Nibhate Jaan Laga Ke
S – Shareef Sirf Maa Baap Ke Aage