Petra is a must but allow yourself plenty of time. It is hard going as there is a lot of walking, its up and down and the paths can be little more than a dirt track.
Go to the Dead Sea and float.
Madaba might be interesting and worth a visit. Its an ancient town which was resettled by Christian Arabs and as such has a strong Christian influence. There is an amazing mosaic map which shows Jerusalem and the Holy Land. Amazing.
Go to the baptism place of Jesus at Wadi Kharar.
You could try a dive if that grabs your fancy at Aqaba.
Amman has an archaeological museum which houses the Dead Sea Scrolls along with other finds. over 9 years ago