I don't know how many days you'll have tickets for, but keep in mind that after 5 days (I think it's 5!), it's only $10 per adult to add another day on. So even if you get to walk around a park a bit, go on a few rides and have dinner in a park, it might be worth adding that extra day. And this is where FP+ is really helpful! You could book FP+ for that arrival day and still get on some favorite rides at that late time!
If you still want to avoid the parks, I think DTD can be a lot of fun. Don't know who your family includes, but there's plenty of shopping! The
Lego store can be lots of fun--they have a building area out back where kids can build and race cars on the ramps. My boys could have spent hours there when they were younger! There's also DisneyQuest, bowling, and movies. There's fun places to eat like T-Rex or Rainforest. And always stop in Ghiardelli's for a free sample chocolate!
Another option is mini golf. The Summer/Winterland course is a lot of fun and if you've got a package you'll get two free coupons to play--just have to pay for anyone else. If you're relying on Disney transport, you can just use the bus to Blizzard Beach and it's in the same parking lot (just need to check operating hours for when you're there).
Also, it depends on where you're staying. Lots of the resorts have fun family activities (trivia, scavenger hunts, sing alongs, etc.), campfires, movies, etc. So there may be plenty to do right at your resort, turn in early, and get a jump on a full day the next morning!