I am quite familiar with LIS as a cruising destination. I concur that Mystic, etc. is an awfully ambitious destination for a 5 day cruise on a 25 foot boat with two kids. If it were me, and obviously this is purely personal preference and opinion, so feel quite free to ignore entirely, I would consider the following:
Day 1: Oyster Bay (about 15 miles). More particularly, West Harbor. It's basically all the way in and around, and West Harbor sits just to the west of Centre Island (the boot). It's great and easy holding, there's a nice beach, and you even can swim right there off the boat in West Harbor. To get to the beach, go to the north end of West Harbor (in that Seawind you can snug up quite close), go ashore, cross the beach that's right there (you can anchor your dink on that beach), cross the road, and you're at a very nice beach. Likewise, there's a little dive of a restaurant on the water there, very accessible by dink, called the Clam Shack. Not at all fancy, but good for kids.
Day 2: Northport (about 6 miles or so from the anchorage in OB). What people call Northport actually is several harbors and towns. If you go all the way into the harbor, that's really Northport Harbor. You can pick up a mooring very easily and get into town easy enough to find a few restaurants, ice cream parlor and a few other things. The moorings there in large part, but not exclusively, are offered by yacht clubs, and they have nice facilities, including pools, for you and the kids to use. Or, if you would prefer to anchor out you could do that too. Before you get to Northport Harbor proper, there's a spit of land called Sand City (that's an informal name known to the locals). You can anchor there and go ashore for another nice beach.
Day 3. Port Jefferson (about 20 miles). Here, particularly if you're looking to go dockside for a day, I'd go to Danford's, all the way at the head of the harbor. (You also could grab a mooring, or drop a hook in numerous places; Port Jeff is an easy harbor with many options.) Danford's is a very nice place, and has showers. You also are right in the heart of town, and Port Jeff has a lot going on, for kids, and all walking distance from the boat. There's a movie theater not too far of a walk. There's also an awesome ice cream shop in the outside mall very close to the harbor.
Day 4. Mamaroneck (on your return trip, about 25 or 30 miles). There is a town float in the West Basin that allows you to tie up to a dock that's floating in the harbor (you still need a dink to get ashore). Once ashore, you are walking distance to Mamaroneck Avenue, which has a plethora of shops, restaurants, ice cream, and stuff. Plus, the harbor is alongside a very nice park. There's also a Brewer's in the East Basin, that usually has a transient slip or two available, in case you'd rather do that.
Day 5. Back to Manhasset (about 5 miles or so), with enough time to go to Louie's for a celebratory drink for you and some chow for the swabs before going home.
That kind of itinerary, in my opinion, still is a little ambitious because it leaves no room for bad weather, or even just to have a day where you're not moving the boat (it's always nice, I think, to be able to chill out for a day or so every few days on a cruise). Likewise, because you need to make miles every day, you will be forced to motor if the wind is not cooperative, either if it's light or if it's on the nose (more likely for your trip home with the prevailing southwesterlies). If you get bad weather or don't want to motor, I would cut out Port Jeff, as that's about 20 miles past Northport, so it's about 10 hours travel time round trip for you, assuming you're making decent way. That said, assuming you want to sail every day and are willing to motor if necessary, Port Jeff is a very doable outside destination for a five-day cruise.
Mystic, however, would be a killer, in my opinion, especially with kids. You are talking about 90 to 100 miles or so from Manhasset. Assuming you average 4 knots, that's about 24 hours of travel each way, not counting the extra time and distance for entering and exiting harbors for stops along the way. Plus, that far east you really do need to mind the tides, as the current out there is material, particularly for smaller boats, which means you may be getting up at odd hours to catch tide, which you would need to do to maintain the pace you'll need if you do a trip like that over only 5 days.
Also to put it in perspective, from an experienced LIS cruiser, we're taking about an 8 or 9 day cruise in August on a 49 foot boat, and we're leaving from Mamaroneck. We're going with two kids (younger than yours), and we're thinking that Block Island might be too much of a push, and BI is not much further than Mystic. We're figuring that we won't go farther than Sag Harbor on this trip, which is closer than Mystic. Now, that's just us, and we prefer not to push particularly long days on the kids because we don't want them to get bored. Our kids are younger than yours, and surely have less stamina for that kind of thing, so I really don't want to throw water on your plans. The above is just one knucklehead's opinion, nothing more.