Japanese American National Museum: Free general admission every Thursday from 5 to 8 pm and every third Thursday of the month. Also free one Saturday each month. This event is called Target Free Family Days and include entertainment, cooking projects and arts and crafts.
Kidspace Museum: Free Family Night is the first Tuesday of every month from 4 pm-8 pm.
LACMA: Free admission on the Second Tuesday each month, free after 5 pm daily and on several Holiday Mondays. LACMA offers free membership for children (and one free adult guest per child). Here is the link for the NextGen membership form.