I found this by asking random moto-taxi drivers what there was to see, so I'm not sure where it is more specifically than a few kilometers outside of town. However they've done an excellent job in preservation, down to the little details of daily life in the structure that used to house and train soldiers to defend the city and the landlords who controlled the town.
Favorite thing: Minqin is part of a huge project to claim chunks of the desert for farming. On the road north you feel you're wandering into the desert, then out of nowhere you're in farm country, immersed in green. Whether it'll last is anyone's guess, but for now fresh vegetables are on the tables in this bit of sand.