We've stayed in both Van Horn and Fort Stockton on two separate trips. Van Horn was a bit better for us -- we stayed at the Ramada Inn there, which had just had some rooms renovated. The pool won't be open, though, as they close it between Labor Day and Memorial Day (or thereabouts). At Fort Stockton, the prices are considerably higher everywhere including at the Motel 6 there.
As glc said, above, Fort Stockton is 120 miles from Van Horn, and almost 2 hours further from Tucson. There are no motels right near I-10 between Van Horn and Fort Stockton; you need to get off the freeway to go into Balmorhea -- only a mile or two, and they're not chains.
BTW, you might be surprised at your little one. If you get off the highway and let the child move a little, about every 2 hours or so, s/he might be okay with 8 hours on the road. BTW, that's about the length of the trip from San Diego to Tucson: right around 8 hours.
In Yuma, you could get off at exit 1, Giss Parkway, and follow the signs to the Yuma Territorial Prison. There is a park there near the river where there is some running space. If you're interested, the prison park has some history and you can do a lot of walking around there. Another place would be Picacho Peak State Park, on I-10 at exit 219. The next day: My personal favorite is the rest area at mile marker 320, between Benson and Willcox.
Donna