Stinson, Castroville, and Boerne Stage are all good choices. Stinson is a towered airport, and has the museum and a restaurant on field (though I haven't been to the current restaurant so I can't comment on it), and is convenient to downtown San Antonio if you plan to spend much time there. The fuel price is higher than the other choices, though, and I can't comment on the FBO since I haven't been there in years.
Boerne Stage is an uncontrolled field with a decent FBO, though I'm biased since we keep our plane there and live very close. It's pretty active, including lots of glider activity especially during the summer. Hertz is nearby, and will supposedly drop off at the airport. Given the location, it may not be the most convenient for getting to your convention, especially with the aforementioned construction on 1604. Since I'm based there, let me know if you have any questions about it and I'll do my best to answer, or give the FBO a call at 830-981-2345.
Castroville is also a nice field (though I haven't been there in a while), and has the best fuel price of the three. Looking at the location of your convention hotel, it may actually be the most convenient choice for driving. I don't know anything about the FBO, though, or if they have rental cars available (if that's a requirement).
Lee