Preheat a grill to medium high.
Mix the butter, tarragon, lemon zest, a large pinch of salt and a few grinds of pepper together in a small bowl.
In a resealable plastic bag, toss the new potatoes with the olive oil, 1 teaspoon salt and a few grinds of pepper. Add the potatoes to the grill and cook, turning once or twice, until well-marked and tender, about 10 minutes.
Serve the potatoes hot, topped with a generous spoonful of tarragon butter (remaining butter can be refrigerated up to 1 week).