Brown left Arsenal at the end of last season to return to the Hammers.
The defender is the son of former Hammers fullback Kenny Brown and the grandson of Irons great Ken Snr.
"I’m absolutely delighted to sign,” the 18-year-old told whufc.com. “This is the Club I’ve grown up supporting my whole life, and it’s amazing to follow the same path as my dad and my grandad.
“Before U9s you can play for as many academy teams as you want, and I did play for West Ham before I signed for Arsenal. It does feel like this was meant to be, and I just can’t wait to kick on now.
“I had a brilliant time at Arsenal, and played under some great coaches, but I’m ready for this next step. West Ham feels like home, and I know a lot of the lads, so I’m just excited to get stuck into it.”
Brown began his junior career with West Ham before joining Arsenal.