For a small state, New Hampshire covers quite a wide range of USDA plant hardiness zones (zones 3b to 6a) which makes it kind of hard to know what will do well for your friend. We could do a better job if we had a more precise location. EDIT -- Sorry, I just saw that she lives near Dartmouth so... pretty cold.
However, there are many crops that love cool weather AND do not require a long growing season. Those would be your best bets -- along with being easy to grow, of
course. I suggest radishes as a first crop because there is NOTHING easier to grow than radishes. The fact that they come up almost immediately (3 to 7 days usually), don't mind cool weather (so you can put them in early in the spring and late in the autumn) and they are ready to eat in about 2 to 3 weeks after they come up, makes them a perfect beginner crop. They also have edible leaves so you get two crops in one!
Other good easy choices for a cool spring are potatoes, onions, beets, carrots, leeks, peas, lettuces, any of the brassicas (cabbages, Brussels sprouts, broccoli), mustard greens, turnips (edible greens as well), pak choy, arugula, cilantro, kale, spinach, Swiss chard, parsley and tons of other greens.
I would also look into some of the early. cold tolerant tomatoes like Siberian, Stupice, Alaska and practically any variety that originated in Russia. You could also try growing peppers and bush cucumbers in pots to take indoors when the weather gets colder.
There really are a lot of good choices that are easy, quick growing and capable of handling a shorter growing season. Just experiment and have fun!