I might have figured out why the contract was not working. I checked my save file further and found a finished tourist mission with another Kerbal named Nelner:
The following happened: I did the tourist mission for Nelner and while doing the tourist mission, I also rescued the Kerbonaut named Nelner from his vessel, and terminated the vessel. So I had a vessel with two Nelners, one tourist, one Kerbonaut. Finally after landing back on Kerbin and recovering the vessel the mission system registered that Nelner had returned, but it only finished the tourist mission, not the rescue mission. Now Kerbonaut Nelner was not anymore in orbit, was not in the astronaut complex, but his rescue contract still existed. I didn't notice at the moment when it happened and kept doing other missions. Until I wanted to finish that mission days later and was stuck with the problem...
Since I finally today realized what happened, I tried fixing it by editing Nelner's name in the save files, but that didn't help (maybe I did it wrong?). Anyhow, I deleted Nelner's rescue contract and now my problem is "solved". But this bug can happen again if two Kerbals have the same name, so my wish is to turn this thread into a bug report, so it will get fixed. I really hope it will get fixed because I got really annoyed, even stopped playing for a few days, and I spent several hours trying to fix it.