I've never been a girl and it's been a decade and a half since I was 16, but below are some thoughts in lower Manhattan that skew a little more to the "hip" side than the fancy side, and won't run you $500. This list will however require you to be around people in their mid-20s.
Buvette - small French place in West Village with terrific food. Really tough to get in for weekend brunch, but otherwise not too bad.
Thai Diner - Nolita is a hip area and this place is great. You might have to wait awhile, but the area is cool to walk around. And if it's too long, just walk over and get in line for Prince Street pizza. Or maybe just do this anyway.
Decoy - underground bougie peking duck place in the west village. I brought my 16 year old niece here and I think I saw her smile over her iphone once or twice.
Cervo's - this honestly may be too adventurous and out of the way (in the lower, lower east side), but it is one of my favorite places in the city and is in a hip area. Again, no reservations, so you may have to wait on a curb with your daughter for 30 minutes while drunk 27 year olds walk around you. If you do this, I suggest walking around the corner for a slice at Scarr's pizza after.