I'd also agree that being near the MK might be best, considering you all have kids (of course, if they're all older kids, that could sway things). I think I'd suggest the Fort Wilderness. First of all, I always feel guilty about spending other people's money, and WL (along with Animal Kingdom) has the least expensive Deluxe Resort rooms. If one family member is treating, that'd help him/her save a couple of bucks. You don't have the monorail, but you DO have ferry access to the magic Kingdom, which is kinda cool, plus boats to the Contemporary, which would get you to the monorail. WL has lake access so you have all the boats to rent, one of the most fun restaurants on property, great pool/kids area...Plus, you're a walk/boat ride away from Fort Wilderness, which has lots of other additional amenities for kids and families, hayrides, campfire singalongs, nightly outdoor movies with the ability to get yerself some 'smore fixins, etc.
To that end, your generous family member might want to consider getting Fort Wilderness cabins. They're comparable in price to some of the less-expensive deluxe rooms, but they sleep 6 (which depending on the number of people in the families, may wind up saving money, as I believe the Deluxe rooms only sleep 4 to 5). They also have full kitchens, so you can make some meals and save some scratch. Plenty of space for the kids to run around, right outside the door as opposed to a long walk and an elevator ride. Trail's End Buffet is one of the best meal deals on property. Pool is bland, but that might be the lone negative. You still get a ferry to the Magic Kingdom.
Those two would be my suggestions.
