Well, the easiest and shortest route from LA to Jefferson City is I-15 north into Utah and then pick up I-70 east to MO. It's also one of the more interesting and scenic combinations of Interstate in the country. You're going to need a minimum of 4 days with a child (how old?). But having multiple drivers will help everyone maintain their sanity and stamina. Plan on stopping every 2-3 hours for at least a half hour to hour out of the car. This can include meal breaks, but should also include stops to hike through some parks or small towns along the way.

The Labor Day weekend should not put a big premium on motel prices, but to be sure, you can surf the web using sites such as hotel-guides and tripadvisor to find prices and reviews. Also note that AAA membership is often good for a 10% discount. You can also consider camping for 1 or 2 nights at state parks along the way - these can be incredible bargains, and will add to the sense of adventure. You will get various advice on this site as to whether it is better to reserve motel rooms ahead or find them as you need them. For my part, I strongly prefer to pre-book, and know the price I'm going to pay and how far I have to get each day rather than start looking at the end of a long day's driving.

As far as that goes, I'd suggest that you make this trip in 4 days of about 450 miles a day. There's a lot to see along the way and such a schedule will allow you some time to stop and at least taste, if not linger over, the best of them. Two good web sites that list off-beat road side attractions are Roadside America and Back on Track. These should get you started on things to see. But don't forego the obvious web searches starting with (for example) 'Utah tourism' and follow the links wherever they take you. Be sure to let your son have some say in the choices of where to stop too.

This should be a fun trip - take it easy and enjoy.

AZBuck