The 18-year-old returned to the Baggies from a loan spell with Northampton earlier this campaign after scoring six times in 14 League Two games.
Bees boss Uwe Rosler told the club's official website: "Saido is an exciting player, who has played for England Under-19s and Under-20s.
"He has been on loan at Northampton and did well at a team that is struggling.
"He is different to Clayton Donaldson and Gary Alexander. He plays in the box, he is sharp and a good finisher. I think he can play with either Gary or Clayton."
