Bowers Museum & Kidseum FREE 1st Sunday every month. 10am-4pm. (Santa Ana)
California African American Museum FREE 1st Sunday every month. 11am-5pm. (Los Angeles)
Heritage Museum of Orange County Santa Ana Residents Only: FREE Admission & Tour 1st Sunday every month.11am-3pm. (Santa Ana)
Irvine Museum always FREE. Open Tuesday-Saturday 11am-5pm. (Irvine)
Japanese American National Museum FREE 2nd Saturday every month except Aug. FREE every Thursday from 5pm-8pm & every 3rd Thursday all day. (Los Angeles)
La Brea Tar Pits Page Museum FREE 1st Tuesday every month. 9:30am-5pm. (Los Angeles)
La Habra Children's Museum FREE 1st Saturday every month. 10am-5pm. (La Habra)
Long Beach Museum of Art FREE every Friday. 11am-5pm. (Long Beach)
Museum of Latin American Art FREE every Sunday. 11am-5pm. (Long Beach)
Orange County Museum of Art FREE 2nd Sunday every month. 11am-5pm. (Newport Beach)
Orange County Museum of Contemporary Art (OCMCA) always FREE Thursday & Sunday 12pm-5pm and Friday & Saturday 12pm-9pm. Monday and Wednesday appointment only. (Santa Ana)
Newport Sports Museum always FREE Monday-Saturday 10am-5pm. (Newport Beach)
Above is a list of some local museums. Many have free days which I've listed. Please be sure to check the provided website links to be sure the information has not changed before driving there. In addition, admission may be free but in some cases there is a charge for parking so take some extra money. And Have Fun!