Now you've really given me an awesome idea. What if the United States annexed most of California, but let the Districts of San Diego and Baja become their own country along with the Southwest Corner of Canon and the Southern Third of San Luis Obispo? I would personally love this, pay for it and maybe even give up Mormon Greenland for it.
Now as for the nation's name, I would suggest San Diego, since San Diego would most likely be its biggest City and Capital. Plus, it would probably be akward to keep the name California. (Personally, I would prefer a San ***** style name to California anyway, despite my secular worldview.

)
And for the nation's culture? It would be the best chance to preserve Califonia's distinct Hispanic culture since TTL's California can't possibly hope to perserve it at the rate things are going.
Best of all, this timeline offers the best oppurtunity for us to have a Realistically Independent Baja California Penisula or an independent San Diego Penisula if my wish comes true.
And just for good measure,
here's California's internal divisions map again.