Some friends and I will be in Boston for a conference next week. On Saturday we are renting a car and will spend the day driving somewhere so I can give them a little taste of my New England home. I can't decide where to go. I want them to see both the wonderful seacoast towns of Northern MA and NH and also the White Mountains. Is there anyway to do that in one day? Or if I am going to limit myself what is the best representation of what NE has to offer?
Any thoughts are appreciated! specific route numbers etc. would be helpful. Thanks!

Choose the mountains or the coast, not both. Since you only have 1 day, I'd choose the coast. Spend the day in Rockport, Newburyport or Portsmouth. Maybe 2 of the 3.

Or if I am going to limit myself what is the best representation of what NE has to offer?

Come back in October during foliage season.

Everything is still bare and brown here, though perhaps White Mtn would be okay due to the evergreens?

Well, I think the White Mountains are beautiful and dramatic even when the trees are bare. And, there are plenty of evergreens.

In winter, and with only one day from Boston, I'd go with Portsmouth, Rockport, Newburyport, also Salem.

Today might have been nice to window shop in Portsmouth but it clouded up after 2pm here to the north of Concord NH. I think your route will depend on the weather. Some roads are already bumpy from frost heaves although the interstates are fine. I don't see any signs of maple sugaring yet but today's warm spell might have started the sap rising. Maple weekend isn't until the end of the month. This week in NH is voting and town meetings.
It might be an adventure to get up early, head straight to the mountains and a late breakfast at Polly's Pancake Parlor. Head back south to Laconia, take Rt 106 to Concord and then Rt 4 to Portsmouth. I always love that approach when you go past Wagon Hill and then over the bridges over Great Bay. Window shop in Portsmouth if weather is good. It would be a lot of time in the car but could be fun. Something a bit closer would be Center Sandwich NH but the Corner House Tavern isn't open for lunch in the winter. The town is beautiful with lovely old houses. Perhaps do some shopping in Meredith, visit the League of NH Crafts. Then head to Portsmouth for dinner. Lake Winnipesaukee is always beautiful. Not sure if they've taken the bob houses off the lake yet, seems like the fishing derbies were just last week.
If the weather is bad, I'd just do Newburyport/Portsmouth etc.