You can stay in the Upper West Side or Upper East Side, which have proximity to museums and on the east side proximity to expensive designer shops on Madison Avenue, but the places to find cute trendy cafes and little boutiques are really the East Village in Manhattan or Brooklyn, neither of which have a lot of hotel offerings.

I'd suggest you find a hotel in Manhattan that you can afford and then simply travel to find the places you want to visit. Manhattan is uniformly safe (at least where there are hotels) and very convenient to tourist sights, restaurants, and public transit as long as you stay between 3rd and 9th Avenues and between 14th and 59th streets. I don't like the area around Times Square because it's so busy, but there are a lot of hotels. I feel the same way about the area around 34th Street/7th Avenue, but there are also a lot of hotels.

You can certainly find trendy areas to stay in, and there are a few hotels on the Lower East Side, which has trendy boutiques, restaurants, and bars. And there are a few hotels there, some of which are near public transit. SoHo is another trendy area, but it's rather expensive.

Most of the museums are on the upper east side, an expensive area with not many hotels. The Surrey and the Mark are both very nice, recently renovated boutique hotels but will be vety pricey. There are some trendy boutiques in the area, but teens would probably prefer a dwontown location like Soho or Greenwich Village. Since the subway and buses will take you wherever you want to go, you might be better staying in Soho and using the subway to get up to the museums. There are chain hotesl like the Hampton Inn, Courtyard and Four Points in Soho as wella s boutiques like the Soho Grand.

While it's true that the UWS and UES side have lots of chain stores these days, there are still plenty of small stores to satisfy a 12 year old. Soho is full of chain stores too, and the boutiques that are there and in other neighborhoods like LES and EV might be out of the interest and price range of a tween. UWS especially is a very family friendly neighborhood and the restaurants are virtually all local and independent, so I still think that's a good choice. Easy to get to east side museums, but a good place to wake up and have a reasonable breakfast or stroll around in the evening comfortably.
Seems trendy enough for most out of town kids and then they can see the local kids out and about too.