I can't help you with places to stay, but here are some ideas about getting there:
The link below will take you to a page with links to several transportation options, including busses to the general area (C & J Trailways, Hampton Shuttle, Greyhound), which should help you find out whether any of them stop in Hampton:
UTS - Getting to UNH by Public Transportation
Another way to get to the general area is by Amtrak, on the Downeaster train that leaves Boston from North Station. They have stops in Exeter, which is next door to Hampton, and on the campus of UNH in Durham. Here is a link that will take you to a page of links to taxicab companies in that area, which should be able to take you from Exeter to Hampton:
Taxis in Exeter - YELLOWPAGES.COM.
Depending on when you're going, it's also possible you might be able to take Amtrak to Durham, and take Portsmouth area transit to Hampton. It looks as if they have very limited service to Hampton, but in case the timing works for you, here is the link to the transit system in the area:
COAST Bus: Serving the Seacoast with Public Transportation
Finally, this link goes to another page with a number of links to transportation options in the area:
GOseacoast - A New Hampshire Seacoast region and South Coast of Maine web site. Check especially closely in the section listing trolleys. It looks as if there may be trolley busses running down the coast from Portsmouth. Depending on where they stop, it's possible that your cheapest option would be Amtrak to Durham, then a trolley bus from Durham, if they stop there, otherwise a trolley bus from Portsmouth after taking local transit from Durham to Portsmouth, or one of the overland busses to Portsmouth and then a trolley bus to Hampton.
Hope this gets you started, and enjoy the concert.