there's really not much "near the zoo, as in easy walking distance. The zoo is almost equidistant from downtown and Mission valley (Hotel Circle). Google map it and pick from numerous well known chains. If you choose downtown, stay south of Market Street.  

There are no hotels near the zoo in Balboa Park. You don't have to drive though. Buses run from 6th and Broadway to Zoo Drive which is a short walk to the Zoo entrance. I recently stayed at the Courtyard Marriott at 6th and Broadway downtown.  While locals might advise against that location I honestly saw no real problems, and I'm pretty picky.