I don't know of any free camping in the bay area. In the Stanislas Nat. Forest (east of the San Joaquin Valley - central valley) you can camp free. You can get a map from the forest service showing all the forest roads with names like 6N27. Some of these roads require a good stout truck and 4x4 isn't a bad idea either. There are many lakes accessable too with good trout fishing.
Father is boss as everyone knows,
But what Mother says always goes.
I have been frequenting Almaden Quicksilver County Park for a while now, which is my recommendation. I usually enter from the McAbee entrance, but the Guadalupe Landfill next door, has started to leak some hazmat of some sort and they closed that section of trail.
So, yesterday I entered from the Mockingbird Hill entrance, which has a picnic area and a restroom (the other entrance does not) and ran out of the parking lot. It is a bit steep in some areas, but what a beautiful trail system. Single tracks that meander through the oaks and cross several seasonal creeks on foot bridges.
I will definitely be returning to this park. It looks to stay cool in summer too with the tree cover.