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Well, I didn't think much of Lanzhou, but I really enjoyed Xi'an ; in fact, it was my favourite city in China. Haven't been to Luoyang though my China travelogue is here. Eliminating Lanzhou is an easy call. Then there is certainly more to see in Xi'An than in Luoyang although you only have one day. It is probably easier to travel in and out of Xi'An too but that depends on the rest of your rip. Xi'an if you dont want to move around.
If in Luoyang, you can head to Kaifeng pretty easily and Kaifeng is worth a day. Actually I already have 1 day in each city. I wanted to know in which city I should spend 2 days instead of 1 only. Lanzhou is just so so? So maybe I could just skip it totally and spend more days somewhere else. Famous for it's prostitutes, supposedly some of the best in the country. Also has a great noodle soup but you can now find it just about everywhere.
Otherwise not much there but snarling traffic and bad pollution. Actually, Lanzhou museum is pretty good and there are one or two nice little parks to wander through If it's a fine day.
You could even take a bit of a cruise up the Yellow River. But, ultimately, given the choice of Xian, Luoyang and Lanzhou for an extra day, Lanzhou would come about eighth. If you happen to be in Lanzhou anyway, there are some pleasant sights with which to kill a day.