The process was run by the Crown Commercial Service and kicked off in January.
The incumbent is Abbott Mead Vickers BBDO, which won the brief in 2014, when it was reported to be worth £10 million. The account was previously handled by Leagas Delaney.
Lida has been charged with developing a "customer-driven strategy" across The Open University’s fundraising activities and attempts to attract more students.
The first piece of work is due to break next month.
Last week, Campaign revealed that Lida had appointed Trefor Thomas, an executive creative director at SapientNitro, as its chief creative officer. Nicky Bullard, Lida’s former executive creative director, is joining MRM Meteorite as its chairman and chief creative officer.
Rapp oversees CRM and data planning for The Open University, while MediaCom handles media.
Debbie Britton, the director of marketing at The Open University, said: "We have been impressed by Lida’s experience, insight and passion for our business."