I'm hoping an LA local will write in with better information, but as someone who flies in and out of LAX all the time, I may be of some use.
The car rental companies are all way off site. (Think several miles.) Once you drop your car off, you will take a shuttle bus to the airport, and the traffic pattern at LAX is as bad as any airport I've ever been to (and we fly alot!) The United terminal is at the end of the shuttle bus run, and those busses have to crawl all the way around an oval pattern to get to your terminal. All by way of saying that you should allow more time than you would think to just process your car drop off. If it were me, I would assume it would take as much as 40 minutes to drop off the car and arrive at the terminal. It could go much faster, but we are better-safe-than-sorry people, so assume as little as 20 minutes to as much as 40 as a good estimate.
If you have a 10:40 flight and want to get to the terminal 1 1/2 to 2 hours ahead of your flight, that means you will definitely be driving through rush-hour, which is no joke in the LA area. Assuming there will be two or more people in your car, most LA freeways do have lanes on the far left for carpools. I know
Disneyland is off the 5 freeway, but I can't remember which freeways you will connect to to get to LAX. (the 10? the 405?) All these freeways, however, are very heavily trafficked and will be clogged during rush hour (often virtually at a standstill).
If there are carpool lanes available, and you use the them, just remember that they can move very fast, then slow suddenly, so stay on your toes. Make sure a pretty confident driver takes the wheel, because you will have to move across 4 lanes or so of traffic when entering the freeway and when you need to get off at your exit. Allow plenty of time when exiting to make this move, since the carpool lanes are marked with broken lines to allow exit and entrance and solid lines when you can't move out of them.
If driving seems a bit more than you'd banked on, would you want to consider returning your car in Anaheim to a local branch of your car rental company (you might be able to find one that doesn't charge drop off fees) and taking the bus? There is a dedicated Disneyland bus service that would pick you up from your hotel and take you directly to LAX for a pretty nominal fee.
If you do drive, and a local doesn't respond here, maybe you could call your hotel to get a good estimate. The last thing we ever want to do in our family is worry about missing a plane, so we tend to spend more time in airports than some people, and we might allow more time than your family would, but I do know that 40 minutes in rush hour will be nowhere near enough time. Sorry if this has been
