Taking a taxi is not an option, because some of the purpose of the trip is to visit sights located in the desert. Also, taking a taxi to Jerusalem would probably be more expensive than driving a rental car; renting a car for five days or seven days is virtually the same cost. Plus, I'm taking a couple of kids that have never seen the Israeli desert (nor any other desert for that matter), and sitting in a taxi with them after a 4PM arrival at Ovda would basically mean that most of the journey would be after sunset.

Because of the late afternoon arrival and departure, I'm thinking that the best option is to stay in Eilat first. On the way back, we could leave Jerusalem at 6AM, and still have time to stop at a couple of sights before we need to be in Eilat at around noon. That is, if we can find a car rental company with a dropbox so that we can return the car on Shabbat.