Don't let them do something that's illegal. Paying for a room for 2 and putting 3 people in a room is actually STEALING. I think you need to tell us what budget they have and dates so we can steer you in the right direction (or tell you it's impossible). There are 2 hotels I know of with 3 single beds in one room:
the Comfort Inn Chelsea
the Wolcott
I think both of these would actually under 21's but you'd have to check. There are also women's residences that are inexpensive and include some meals. The Brandon is one of the better ones. http://www.thebrandon.org/Content/25/SC1001.asp

Great ideas! Thank you.

Ah. Memories of my first trip to NYC as a college sophomore and staying at the West Side Y for $6 a night -- nearly a hundred years ago, LOL. I thought it was still considered a good bargain place, private rooms but mostly shared baths. I do know they now have "triple" rooms. It's location right off Central Park at 63rd was great. Anyone know more about it?

It's still there and still good. I mentioned it up top as one of 2 well located hostels for them.

If sleeping on the floor is the worst thing she ever does in her life, than all cocerned would be lucky.

That is not a law, it is a means to icnrease business. I would not think twice about it and have not in the past.

Aduchamp1 , the concern is not "sleeping on the floor" (but--yuck) but exceeding the limit according to the fire code.
And of course some of the rooms are so small, you literally couldn't fit an air mattress on the empty floor space.

they can book at triple with or without a shared bath at the carlton arms for $140/night. It is an artist owned converted SRO that is popular with Europeans. They will rent to 18+. I stayed there many a year ago on my 1st trip to NYC prior to moving here a month after I graduated High School...
http://www.carltonarms.com/

It is very very convenient to going out/getting around.