The area that makes the most sense would be the MS Gulf Coast, Biloxi-area, which should be around half way from Lafayette.
Unless you want regular shopping, chain restaurants, and stuff I think you'd have to dip down off 10 to Hwy 90 for most things. The Stennis Space Center is right off 10, but I've never toured it and don't know what they offer as far as tours and sights and such. I don't know of much that would necessarily be greatly entertaining for kids, but they do have the Ohr-O'Keefe Museum in Biloxi that was designed by Frank Gehry so the architecture of the building itself is worth seeing. Beauvoir, the Jefferson Davis home is west of Biloxi on 90.
Half Shell Oyster House would be good for lunch, or maybe one of the Casino buffets, if you're into that. Just east over the bridge is Ocean Springs which has a neat historic downtown with restaurants and shops to walk around. The best restaurant with the best fish sandwich on the entire Gulf Coast closed, unfortunately. The Tato-Nut Donut shop is there and is kinda renowned locally.
The Abita Brewpub in Abita Springs might be a good place to stop for lunch, and there's a little town center with a museum and a park with a splash pad.
Bozo's Grocery in Pascagoula has some of the best Po-Boys around, another lunch option.
Mobile has the USS Alabama and the Gulf Coast Exploreum Children's Museum.
Pensacola obviously has some cool things to do like the Naval Aviation Museum or maybe even a Blue Wahoos game, but they only have day games very rarely and it's hot.