Hi S.,
While Monterey Bay Aquarium is a great place to go with kids, even young ones, their kid specific area "Splash Zone" is currently closed while they completely redo it. It should be even more amazing when they finish. I would recommend keeping an eye on their website for progress and reopening information.
As has already been stated, Happy Hollow Park and Zoo in San Jose is great and much of it is very much geared for the toddler set. It too will be undergoing some major renovations and rebuilding and closed for about a year, however I believe it is still open at this time. There is a parking fee on most weekends. If your family like it and want to go regularly I strongly recommend a membership and they also sell parking permits (which are good for all San Jose City parks).
I also agree with those who recommend Children's Discovery Museum of San Jose, it has some great play areas and a toddler specific room. There is another Discovery Museum in Marin? that is also fun. There is a level of membership you can purchase that will allow entrance for both (and others).
The free magazine "Bay Area Parent" often has articles or advertisments that will help you find other appropriate places, and their calendar can be very helpful for ideas of what specific events are coming up. They have an online version too though I haven't checked it out yet.
While I think it is great your son wants to go adventuring with his son, don't forget the value of some quiet, quality time together (playing, reading, napping, singing, sitting in the bath together having a good splashy time, and all those other wonderful, simple activities that really help a child grow and develop, while also potentially building a great relationship).
What ever he decides, I hope all have fun.
K. H.