Is there a train from Boston to Meredith NH?

Unfortunately, there is not Amtrak service in New Hampshire except along the Atlantic Coast. White River Junction, VT is the closest train station to Meredith. You would have to take the Lake Shore Limited from Boston to Springfield and change to the Vermonter to White River Junction.

Your best bet is probably taking the Concord Coach from Logan Airport or the South Station to Meredith. There are two buses a day each way.