San Diego Cheap Eats
You have lots of great options in that location.
Mona Lisa Delicatessen - the restaurant is lame, but the deli is great. This place is in Little Italy, and has very good sandwiches and Italian deli salads. Very inexpensive, and the sandwiches are made w/ high quality ingredients.
Pete's Meats - also in Little Italy. Great sandwiches w/ freshly grilled meats (or eggplant). Meatball, sausage, spitini, and steak are all worth getting.
Mimmo's Italian Village - yet another Little Italy spot. Good sandwiches on good bread, and also Italian deli salads. I'd avoid their hot pasta dishes, they are not so good.
Blue Water Grill - up India St. from Little Italy, about 5 miles or so. Fresh fish, prepared in a variety of ways. Sandwiches, salads, or plates. Standard menu items are all under $20 for a plate, special fresh fish options sometimes creep over $20. Great stuff.
Saffron - quasi-Thai food on India St., a block up from Blue Water. This place has a lot of detractors, and in some cases I agree. The chicken breast skewers are horribly dry and bland. Too many of the curries made w/ white meat chicken also suffer from dried out meat. However, the soups, rotisserie chicken parts, and spicy noodle dishes are all decent. The rotisserie chicken dinners are cooked on the bone, as whole chickens, so are juicy and flavorful. The cold salads they come with are also very good, and some of the homemade sauces are great (particularly the Sriracha and the peanut).
El Cuervo Taco Shop - this is up the hill from Little Italy, in Hillcrest/Mission Hills. The things to get here are either the carnitas or the al pastor. It's very cheap, and the food is tasty, albeit nothing fancy. Some of the better carnitas in town, and the best close to where you're staying.
Good luck on your quest!
You're only minutes from Old Town. It's been a few years, and I know Casa de Pico has relocated to Grossmont, but there have to be sub-$20 meals to be had. Maybe Casa de Bandini? You'll pay extra for the margaritas, but it's a fun atmosphere.
Point Loma Seafoods is another option (2805 Emerson St., Pt. Loma). Nothing fancy but the fish is fresh.
As for lunch, almost anyplace will get you an entree less than $20. I know some will disagree, but whenever I'm in town I head to South Beach Bar & Grill in Ocean Beach. Great fish tacos in many varieties (mahi mahi, oyster, lobster, etc.). Plus the view of the O.B. pier (and the locals) is definitely memorable.