Yeah, National Forests and certain wilderness areas are probably the way to go for overnights. The main issue is that some dogs harass the wildlife. Also they don't follow LNT practices, not even German shepherds, and they are smart! They do UL just fine though.
Best place this time of year may be south of Monterrey. Since horses are usually allowed I assume dogs would be ok. Check first though. Signs are this year may be a bigger than usual tick season, and that area is already known for its tick population, so plan accordingly. That area is really beautiful in March and April. 2 to 3 hours at most from most places in the bay area (17 to Hy 1) south, or 101 south to enter from the east side.
You could check out the book "Hiking the Big Sur Country: The Ventana Wilderness" by Jeffrey Schiller. Bit old but pretty good for some decent starting ideas in that area. Ventana has lakes and hot springs as well as rugged mountains and rivers.
I did the Ohlone Wilderness Trail a few years back. I didn't know dogs where allowed, but if so that would also be especially nice this time of year as well. Camping only in designated spots however. No lakes (a cattle pond or two) but there is a waterfall.
Edit: Ah, OK I see the footnote from a previous post that Ohlone is day dog only. Not a bad place to day hike though.
Unfortunately the immediate bay area may have the worst combination of easy access great places to hike, and worst places for backpacking, especially with dogs in the US. You would be hard pressed to find many places that would be considered in the bay area proper that will allow dogs away from the entrance/car camp areas, and probably none that would allow them off leash. Probably a good rule since it would most likely ruin the parks with such a high population. Since the sierra national forests are a bit far for weekend trips, I would say that probably leaves the coastal range in the Big Sur area as one of the few places for weekend overnights w/dog.
Wow, for some reason I have been obsessed by this issue for the past hour! I love dogs but don't have one anymore. I never though about dog backpacking before. Anyway, going to move onto my next obsession. LOL. However, have you seen this web page:
http://www.naturedogs.com/findahike.php