You Have Time to Burn, but Money?
The first thing you need to recognize is that 10 days will let you get anywhere in the contiguous United States as well as most of Canada, so there's simply no way that we can offer any meaningful suggestions. That first choice - of destination - is entirely up to you. But I will say this about your basic plan: you have chosen the two most expensive options for travel that there are, a one-way rental, and one-way plane tickets. If you are under 25 then you have to add another $250 for each of you who plans to do any driving during your trip. If you are all under 21, you can simply forget such a rent/fly trip for the time being.
Now, how can you save some money. You can, in fact, save the entire rental fee and the cost of the plane tickets by simply using one of your own cars and doing a loop trip. From Boston in winter, I would think that the South would hold a certain attraction, not the least of which is beaches warm enough to go swimming. A loosely defined loop might look something like drive the length of the Jersey shore, take the ferry over to Lewes DE, continue down the Delmarva Peninsula, cross to Norfolk and then explore the Outer Banks of North Carolina, take another ferry back to the mainland at Cedar Island, check out Charleston and Savannah, hit Okefenokee Swamp, cut across to the Gulf Coast and explore that to New Orleans (while in the South, you might even be in the right place/time to catch a bowl game), then start for home by heading up the Blues Highway to Natchez and picking up the Natchez Trace Parkway to Nashville, continue across Tennessee to the Great Smoky Mountains, and then follow the spine of the Appalachians back up to the Berkshires and home.
A trip like that would let you see a ton of country, with lots of great scenery and a host of different cities for not much cost.
AZBuck