I think this has "trouble" written all over it. First issues: Safety. She's a completely inexperienced driver, but she wants to have a heavy machine that she can physically control as well as drive (2 separate issues) and which offers her zero protection or safety (no seat belt, no air bag, no protective layer of anything around her). She will be a target for bad auto drivers out there. And she wants to share this with someone else, which means they will be riding together - so on top of the inexperience , she's going to ride with the added weight and distraction of this guy sitting behind her, or she's going to ride behind him.
Second issue: practicality. She can only ride it in good weather, which in NH is April through September, if there's no rain. She can't carry a lot of stuff on it - a backpack full of books for school? Summer recreation stuff? So if she's telling you what a convenience it would be for her to have her own vehicle, she's dreaming. Most kids promise that, if they can get a car, they will do errands for Mom and Dad, go grocery shopping, take the siblings to friends' houses, etc., and they don't follow through - but she's not going to be able to promise those things with a motorcycle! Then she has to negotiate using it with another person? So she wants to ride to her job and he wants to ride to his, and the bike sits in the parking lot for hours? They're going to be fighting over this all the time. Who wins? Who decides? What does the one do when the other has the bike?
Third issue: expenses. She wants to go into a business relationship with someone she knows, but she is a minor and maybe he is too. They cannot sign contracts, and so this will break down into arguments and disputes. I'd have her go through the exercise of "what it really costs" which means the price of the bike, registration/licensing fees (for the bike), gas, repairs & maintenance costs, helmet, goggles, protective clothing, and the outrageous insurance costs for 2 underage drivers on a motorcycle. They will each need their own policy including collision, liability, etc. Have HER go through the exercise of calling 2 or 3 insurance agents to find out. Do not agree to put her on your policy. Then have her research the cost of getting a driver's license and then a motorcycle license on top of that. She wants to "learn" how to ride a motorcycle? How is she going to do that. Does the local driving school offer driver's ed on that? If not, is her teacher going to be some kid who had a bike for a while?
Remember, and remind her, that a young person's brain isn't fully developed until around age 25. The last part of the brain to develop is the part that predicts consequences. This is part of the reason why so many young people are in car accidents - no experience mixed with poor judgment and the inability to anticipate.
My son talked about this for a while. We said no. He's pretty thrilled that we did now that he looks back and sees what it would have been like. After he had his first car accident (which wasn't even serious in its consequences although it could have been - he just skidded on icy roads because he didn't slow down enough for the existing conditions and couldn't get the car out), he realized we were right. He wound up getting his own car through his grandmother - a gift he was lucky to get - and later saved up to buy his own new car after he got out of college. That's the way to do it.
Once she's out on her own, working full time and paying for an apartment and health insurance and groceries, she can do what she wants. If she's under your roof, you have to put your foot down. But I'd absolutely take her through the exercise of planning a budget (her income vs. all her expenses) and researching all the hidden costs involved. My guess is, she and this boy have looked at the purchase price of the bike and maybe a helmet, and that's it.