When is the best time to plant a mophead? I live in Southern CA where the climate is mild to hot year round.
Plant whichever weeks your temperatures are coolest and there is less heat and water stress on the shrub. In your climate, I think you'll want a site with filtered to partial sun, planted facing east if possible where it's not in the strong afternoon light.
Good luck growing hydrangeas in Southern California!
Hydrangeas grow beautifully in SoCal! Coastal SoCal is better suited than the hotter/drier interior areas but otherwise, very few limitations as to even the best time for planting. Whenever!! There are some issues to keep in mind - shade during the hottest part of the day and perhaps a need to create a specialized planting area with pH-adjusted soil. And if you have issues in your area with drought and the municipal encouragement of xeric or waterwise plants, you might want to rethink hydrangeas altogether.
My sister lives just south of Laguna Beach and has a quite shady courtyard garden that has a bunch of hydrangeas, both planted and in containers. They slow down a bit in winter but otherwise never seem to go completely dormant or lose all their foliage and can produce blooms at nearly any time of the year. She often comes home with some new florist's hydrangea "find" she picked up at Trader Joe's or the grocery and after a week or two indoors, she plants them out as well.
Sunset's Western Garden Book recommends mopheads (Hydrangea macrophylla) and most other species for all California climate zones except the high desert area of interior Southern California (Barstow, Victorville, Palm Springs, etc). If you can grow hydrangeas in Texas, you can for sure grow them in most of California!