Huhhh, interesting.
Since people seem to know stuff here:
I had a mission last night where I was taking some quest item propaganda somewhere and someone wanted to pay me more for jettisoning it. They said they were gonna make it worth my while if I went to system X. So I do that, and nothing happens. How do I find what happened?
You look into the mission detail in your transaction tab if it shows you another mission. In your case, I think you need to abandon the mission and then just jettison the cargo. It should then point you to the station (usually the one you took the mission from) you can get your pay from.