One of our recent free finds is the Santa Susanna Train Museum and adjacent park. It's not technically in LA, I guess, but right next door in Simi Valley. Since we live about 3 freeway exits away, it's definitely close enough for us. It's an old railroad station that was moved and turned into a small train museum and model train display. The museum only has a couple of rooms but you can blow a train whistle, play with a telegraph and teletype machine and see old train-related paraphernalia. And get a sticker, which is a big hit with the 4 year old crowd.
The highlight for us is definitely the very large model railroad exhibit which has a very active group of railroad enthusiasts (very enthused enthusiasts!) who take the model railroad extremely seriously. Apparently they occasionally have get togethers where they run the trains on a schedule and move freight around. The railroad is fashioned after the California coast, which is also kind of neat.