Close to Times Square: A fun and inexpensive place for teens is Vinyl...cool decor and decent food (9th Ave and 51st.) John's Pizza is great, the original is on Bleeker St in the Village but the one near TS is in an old church, much larger and prettier.
For Jewish deli we love the 2nd Ave Deli (on 3rd and 33rd) but many prefer Katz in the Lower East Side. Known for the funny scene from When Harry Met Sally so mobbed with tourists on the weekends.
Many, many great Italian places....we have enjoyed Babbo, Maialano, and Locanda Verde recently.
A great resource for dining recommendations is Chowhound. They have forums like this totally dedicated to food by area. Check the Manhattan board.
If you are looking for recs on things to do with teens, I love the Tenement Museum, also in the LES. Great small tours of an actual tenement building restored to different time periods. Should probably book in advances and the group size is limited. On our last trip we visited the Frick for the first time. Great small art museum in an amazing setting (the actual Frick home.)
If you are there for a week I would definitely try to get to Brooklyn too. Williamsburg is the center of hipster cool and the Brooklyn Flea and Smorgasburg are great fun. The Transit Museum on the Brooklyn side of the Brooklyn Bridge is very interactive, and is easily combined with a walk across (and the River Cafe, if you are so inclined.)