From SA myself.
Hiking? Enchanted Rock, if you are renting a car. Beautiful hiking especially during this time of year. If you're gonna head out that way, you can always check out Lyndon B Johnson's ranch and Hill Country towns like Fredericksburg in the area. Anywhere in the Hill Country is amazing for hiking purposes.
How old is your son? Sea World or Six Flags Fiesta Texas is a fun option if he's a bit younger.
While the Alamo is the most famous Mission, there are others out there that are more beautiful. Try hiking the Mission Trail to capture all of the missions and not just the Alamo.
Breakfast tacos are a must when in SA. Tacos and Salsa is probably the best place for them. Other than that, you can't really go wrong with any Tex-Mex restaurant outside of downtown/touristy spots. If you don't serve good Tex-Mex in SA, you're not gonna last long as a restaurant.
The Pearl is really nice especially at night. A few Michelin star restaurants are available there as well, if culinary arts are your thing.
You'll also be in town for the world-famous San Antonio Stock Show and Rodeo, one of the most premier rodeos, with star-studded lineups performing. You can always check that out and explore the fairgrounds. They've won best rodeo a few years running. It's a month-long city-wide celebration.
Unfortunately, because the Rodeo is in town and takes over pretty much everything, the Spurs won't be playing in town as they'll be on their annual month-long "Rodeo Road Trip" and won't play back in San Antonio until March.
