In Leicester Square (not the tube station, but in the actual square), you can find a lot of restaurants and places to catch a bite to eat.
A nice and reasonable bar/restaurant is The Moon Under Water, on one of the corners of the square. They have all kinds of food for good prices and drinks as well. Just around the corner from there is Wagamama. It's a Japanese canteen-style restaurant, and the menu is extensive. Good food, and the prices aren't that bad for being London.
On the left side of The Moon Under Water, there are some other pubs and wine bars. Quite pricey, though. Beside them is a very nice Mexican restaurant. I would recommend reserving a table if you're planning to go there on a Friday or Saturday evening. The place is a bit American style, but the food is yummy, although sometimes you have to wait a while for it, since the dining place is big with many customers. Then, of course, there's a Burger King in the square, a Pizza Hut, a Haagen Dazs café, which is gorgeous(!), and some other small places. To the left of the Odeon cinema in the square, there's a small road which will lead you a place called Waxy O'Connors. This pub used to be an old church, and the architecture and layout is spectacular. They have lovely food as well as many, many drinks. On Friday and the weekends, this place gets very busy, but all in good spirit! I highly recommend this place.
In conclusion, Leicester Square will lead you to many places around the area, including Soho, Chinatown, and Covent Garden. It's a good place to just walk around and explore without a map and see where you'll end up. Good luck!