If you just want to walk around the suburbs of NYC go in the other direction and hop on a train to Princeton, NJ. If you get an express train out of Penn Station it is less than an hour. Tons to do there including the university and some decent restaurants. If it’s a nice day weather wise my favorites would be the outdoor patio behind the Nassau Inn, bar at Agricola for a burger/beer or communal tables next to the bar at Mediterra. That would be a nice day if you have good weather and very easily done as a day trip out of Manhattan.
Mystic is lovely, but I don’t think it’s worth going many hours from NYC. My vote is Hudson Valley. Or maybe a boar ride up the Hudson River. (If you take the train, you also get a great river view)…
Mystic is absolutely marvelous if you can spend an overnight, but a day trip will not allow you to see the town adequately, since the train ride is long.
Do something different that doesn’t require a great deal of travel- take a ferry from Brooklyn Pier 6 to Governor’s island, either rent a bike or walk around. You’ll have a very different experience of the city while never really leaving it.
I agree skip Long Island . I grew up there. Your time is better spent going to other areas . I agree with Princeton or Greenwich. Lovely towns and easy to get to by train.
Greenwich is fun. Get off at train station and walk straight up (it’s a hill) Greenwich Avenue to shops, if you haven’t gotten fed up with them in NYC. Hermès and Saks are closer to the train station, but further up on the same side you have to check out Betteridge (absolutely gorgeous jewelry, fun to window shop), and on nearby Lewis Ave Stephen Fox jewelers.
Or if you’re not into seeing the shops, the Bruce Museum is right in the same area as the train station as well. Free admission on Tuesdays.
Just as a technical point, Brooklyn and Queens ARE part of Long Island.
Are you trying to start a war? 
I vote for Tarrytown. There is a big castle there. Sleepy hollow is right there. And right next to that is the rockefller estate and blue stone barn (i think that is the name) which is a big farm/ self sustaining restaurant.
If you dont want to go far, Bronxville in Wesrchester (Metro North RR) is very cute. Sarah Lawrence is there.
Blue Hill at Stone Barns has fabulous food, but while it’s only a little over three miles from Tarrytown you can’t get there with public transportation. You could also go to Equus in Tarrytown. It’s in a Castle with great (distant) views of the Hudson. And I worked on the interiors many moons ago! (Though I’m pretty sure all those carpets and fabrics have probably been replaced.) I haven’t eaten there with the current chef.
Just pointing out that she could go walk the promenade in Brooklyn Heights and say she’s been to Long Island 