Akindayini was released by Spurs at the end of June, despite a promising campaign for the under-21 side.
His departure was on the cards from late April, after Ipswich Town's website confirmed he was on trial with the club.
Brighton's website report Akindayini will begin training with their under-21 squad rather than the first team, with the 19-year-old still considered a player with raw potential rather than a proven commodity.
Sealing a move to the Championship side is in itself a good nod to Akindayini's ability, with another player released by Spurs, Alexander McQueen, moving to League Two side Carlisle United.
Also released by Tottenham were Jordan Archer, Jonathan Miles, Bongani Khumalo, and Cristian Ceballos.