The appropriate way to do Storyland, I've heard, is to jump the fence at night, but that's probably not Lukas-appropriate.
Assuming you're staying nearby … (this map
will help, as would the White Mountain Guide)
There aren't a lot of roads which are nice to run and flat (or flat-ish) that don't have a lot of traffic, especially during peak leaf peeper season. The road from Route 302 up towards Rocky Branch (Jericho Road) might work? Hurricane Mountain Road is super steep, but traffic is slow/minimal. Some of the back roads to Jackson (Dundee, Thorn Hill) are nice, too, but if you're going to go on that much of a hill, just go on a trail.
Okay, so trails. There are lots of these. I'll assume you want to drive no more than 20m from Storyland.
Just east of NoCo there are some trails in the Green Hills Preserve which are reasonably short and have excellent views. Peaked Mountain, for instance. They might not even be too overrun by tourists. But they might.
Just west of NoCo are the Moat Mountains, which are pretty proper trails (I've never actually hiked/run them) and more by cathedral ledges. The Moat Mountain-Red Ridge loop is a 9 mile loop with considerable elevation.
Kearsarge North (off of Hurricane Mountain Road) is 3 miles up, but with considerable climb, to the summit of Kearsarge, which has excellent views.
Up in Jackson, Doublehead is a nice climb and has decent views on top. The Hall's Ledge trail up at the end of Carter Notch Road is a nice run on ski trails to a viewpoint; there are some nice loops/out-and-backs you could make on the ski trails in Prospect Farm. The Hall Trail off of Iron Mountain Road is a nice dirt road to run.
Out the Kanc, there are several options for shorter hikes, like the Mount Potash Trail (4 mi RT) or the UNH trail (5 mile loop). Longer trips up Mount Chocorua, which will probably be teeming with people.
Up Crawford Notch is Mount Tremont or Mount Crawford off the Davis path. Both longer trips.
And then you can head up towards Pinkham with all the trails there. A run up to the base of Tuckerman and back is always nice, but if you go that far, it's hard not to keep going up the mountain. Plenty of options on both sides, especially if you have time and the weather is nice.