We go there a lot (got family there).
Trails – you’ll need to travel to get anything good.
There are a number of small canyon and ridge sections close to the coast (Tecolote Canyon, Rose Canyon, Los Penasquitos) ok for cruising but nothing challenging, and there’s a small trail park behind UCSD that can give an hour of fun. Mission Trails Park a few miles inland has quite a bit. The locals are a bit protective as there have been access issues !
Nearest decent is an hour inland at Mount Laguna, which has quite a bit of trail, some quite benign, but Noble Canyon is right up there http://www.singletracks.com/bike-trails/noble-canyon.html , ideally done with a vehicle at each end. There’s some decent riding near there around Cuyamaca too.
Travelling a bit further there’s good stuff in Idyllwild https://youtu.be/sW0auLo5bpg and some other locations – I’ll ask the boy for some more ideas. I’ve never had to rent as I use his bikes but SDMBA has some links.
And btw, there’s a good network of bike trails for safe family riding round the bay, seafront, etc.
(posted same time as entry above !)