Malibu is a sleepy little beach town with few hotel and restaurant options. Therefore we recommend staying in Santa Monica. (See TripAdvisor recommendations.) The Ambrose Hotel (see 1 below) is a very nice, somewhat affordable hotel in Santa Monica, where we stayed the first time we visited.
Santa Monica lodging can get a little expensive during the touristy summer season. A less expensive lodging option is to stay in Marina del Rey, which is just 3 miles to the south of Santa Monica and very close to the airport. We've never stayed at these hotels, but here are a few that rated high on Trip Advisor's list and are reasonably priced. Each of these pages has a link that will allow you to book a room through Expedia and other websites.