Did you bind off any part of the neckline before picking up stitches for the neckband? Those stitches also need to be bound off very loosly.
It may be that even with your best efforts to keep your bind off tension loose, you still may have bound off too tightly. It may also be that you really need to knit the neckband on a different size needle. I think for a better look, it'd be best to rip out the neckband, reknit it on the next larger size needle, baste the seam closed when you are done and before binding off, move all stitches to a scrap piece of yarn and try on over his head. If it fits that way, then you know that you simply need to make your bind off loose enough. If it doesn't fit then the neck hole really isn't large enough, and you may need to rip back a little bit of the body to fix that up.