A few weeks back, my family visited the Banning Museum, not far from the docks at San Pedro. The museum is comprised mostly of the mansion of Phineas Banning, who was instrumental in delivering people and goods by stagecoach all across the Southland in California’s early days. Parking is free, and so is access to the grounds (including a picturesque rose and wisteria garden), but there is a small fee for tours of the house and a new exhibit on travel to and the settlement of L.A. County. Our tour guide was knowledgeable and friendly, handling my children (ages 5 and up) expertly. Kids will enjoy playing at the adjacent playground before or after their visit to the mansion.
The museum website is not very helpful (one of the docents informed me it will soon be updated), except that it suggests to phone (310) 548-7777 for more info, so I highly recommend phoning before scheduling your visit. The wonderful volunteers will fill you in on tour times and prices.