The Lille midfielder looked set for a move to east London after the club agreed an initial £29m deal, although the switch broke down due as the two parties failed to agree personal terms.
Onana was in attendance to watch Everton lose to Chelsea on Saturday and has since sealed his move to Goodison Park, with Lampard predicting a big future for the 20-year-old.
A Belgian international, Onana had only been at Lille for a season after impressing with Hamburg and has now signed a five-year contract on Merseyside, joining another West Ham target in Conor Coady there.
When asked of his decision to move to Everton, Onana said: ““The manager was one of the big reasons.
“He played at the highest level, won many things and was a midfielder, too. It means a lot to have interest from him and I think he can teach me many things.”
Having missed out on Onana, West Ham are now trying to strengthen with other targets and are eyeing up to four more signings between now and the end of the window.