The key to having fun in desert driving is not too heavy a vehicle (as you want it to float on the sand), a big engine (4 litres or close to it), a short-ish wheelbase, and a real 4x4 drive system, with Differential Lock if you can get it. My three choices below all adhere to this.
For that price, the only options I would consider is a Jeep Wrangler, an FJ Cruiser, or 3-door Pujero. I know that right now, there is for sale on the Abu Dhabi 4x4 club website a VERY nice used 2009 model year Wrangler with all appropriate offroad mods that has been used by one of the Marshals of the group, for a bit over 60k. So a bit higher than your budget, but if you can stretch it would be well worth it - I've seen him drive it and it will go pretty much anywhere you have the nerve to drive it. (http://www.ad4x4.com/club/viewtopic.php?f=3&t=22154&sid=cb76ec2e6c8ce9675a67cdb7fafb249f)
If you want to use it on the highway as much as the desert (my vehicle is both a Daily Driver and a Weekend Warrior), then look for an FJ Cruiser - more comfortable and safer on the road than the Wrangler, much more survivable in a highway accident (but to be honest, not as good on the dunes as a fully upgraded Wrangler). But Toyotas really hold their value, and I fear that you might have to go back a few more years of age to get a similar price to a Jeep.
You can go for a 3-door short wheelbase Pujero as well...the previous generation Pujeros are pretty nice, and in some ways better than the current generation, as they lack a lot of the plastic bits on the bumper that have to either be guarded or cut away to make the current Pujeros usable in the sand. Again, they hold their value, and not as good off-road as an FJ IMHO...so why not just get an FJ in my thinking? But if you can find a good used 3-door one, it will do the job.
That is pretty much ALL I would consider, really. The short wheelbase Nissan Patrols are underpowered in my opinion, the long wheelbase Patrols excellent but would have to be very used to fit your budget. Anything that says "Rover" would be either too expensive (RR, RR Sport, LR3, LR4), unsuited (LR2), or too high a center of gravity (Defender) to really have fun with in the bowls and switchbacks. The other Jeeps are not as good in the sand as a Wrangler, nor as cheap. Besides Jeeps, the only American 4x4s that are fun in the desert are F-150 Raptors, but waaaay too expensive. The other American SUVs are basically jacked-up trucks with poor 4x4 systems, except Hummers, which are just WAY too heavy to do any real freestyle driving with (they WILL take you across the dunes...you will just be bored).
Mind you, I was out in the desert last weekend and we saw some Korean-SUV that someone had put a HUGE lift kit on and was out tearing up the dunes...a lot of dune driving is the driver, not the vehicle. Oh, and your recovery kit...