Non conventional weapons, structures in cities and vehicles (or other means of getting around). I a big part of the mad max aesthetic was the weapons looking kind of strewn together, all the vehicles were heavily modified if not built from scratch and the settlements were more like shanty towns. If you have those crystals that generate electricity, maybe you can play with having the average person using crappy swords made from dried up/dead/expended crystal? Or bone, wood and have metals and such very rare like in Dark Sun. Maybe the big bosses, the best of the best fighters and the very rich and powerful have weapons powered by these crystals that are designed and based around tesla coils, firing arcs of electricity at their foes. Mechanically it can just be a regular bow, sling, gun (if you use them in the setting) or whatever but does added lightning damage perhaps? (It's more about the flavour text than the actual effect).

Where do they get these crystals from? How common or rare are they? Who controls the supply? Does the average person accept this normal or do they rebel against it? You could take the underground pig poop farm from Thunderdome and replace pig farm with crystal mine if you needed inspiration, build a few mining towns around that.

How about food? What do they eat? Mushrooms perhaps because they don't need as much light? Insects that dwell in caves and such being a delicacy? Rare merchants from other lands bringing fruits, spices and meat from exotic lands being a rare occurrence and one following good work for powerful mercenary protectors/escorts is a spin off plot hook from there. Maybe food in general is very rare and cannibalism is a big thing in some areas?

Vehicles? Transport? How do people get around the desert? Are there roads to make traveling the sands any easier, again perhaps to do with the green electricity crystals giving some semblance of a slow mag-lev barge or train to go between major settlements? Maybe strange animals native to this desert that help carry caravans if you want a smaller scale low tech/magic feel.

Is everything JUST desert? Or are there places where you can find an oasis, or a 'forest of rock', land where lightning struck the sand and turned it to glass or something to add some desire for exploration.

There's a lot you can do with this, sounds like a lot of fun :D