A couple of thoughts:
1. It sounds like (if you have the whole story) she's asking for this for the right reasons. I'd shxt my pants if my unwed Mom wanted me to uproot at age 16 from one hick town to another, to live with someone we'd never met. Chances are, that other "girl" is a nekkid obese 45 yr old pervert looking for a chance to "move in" with a mom and her 16 yr old daughter. Ugh. Run the f**k away girl, smart choice. Get out by any means possible.
So... allowing her to move in with you might prevent a runaway situation anyhow.
2. You'll be taking on a pretty big obligation. 16 year old girls get taken advantage of no matter where they are, and you and your wife will be both her mentors and protectors. Then again, you're an NCO who's seen this sort of thing before, the only diff here being that your niece hasn't signed a contract and isn't drawing a paycheck in return for getting up every morning at reville.
3. She's probably spooked by the decision her mom is making and by her previous situation, so you'll need to avoid coming across as too authoritarian when you discuss the household ground rules with her. She's old enough to want/expect some independence, and you'll need to walk the lines between friend and parent, and good cop / bad cop.
I guess it comes down to if you think your household is healthy enough to absorb an unknown factor and thrive. Still, the girl's situation sounds like a bad cable tv movie waiting to happen if she doesn't have the option to stay with you. I don't see much good coming from telling her she has to stay with Mom and whoever it is she's moving in with, in a small town that won't help her meet her real potential.