The 21-year-old, who has enjoyed a successful trial period at St Andrew's, arrives from League Two strugglers Barnet.
A former West Ham United trainee, Lee - son of former England and Newcastle midfielder Rob - enjoyed loan spells at Dagenham and Gillingham before signing for the Bees last summer.
Blues boss Lee Clark told the club's official website: "We have been scouting Olly for a while now and he was one of Barnet's stand-out performers during our Capital One Cup tie earlier this season.
"Olly has impressed us further during a recent trial period and we now look forward to seeing how his development continues over the coming months."
- Sports & Recreation