Jenkins Pipeline job - remember previous values
Depending on how long you keep builds for, the Rebuild option in the left hand sidebar will allow you rebuild a previous run of the job with the parameters intact. 

Simply find a previous build of the job that that particular user kicked off last time, click into it, then choose Rebuild from the sidebar. 

Again this will only work if Jenkins keeps enough builds. We save two weeks for a particular job and I have a coworker who saves a link directly to x build of the job, then once he rebuilds updates his link to the new one, etc. This way he can just open the link he has saved, click Rebuild, and he's good to go as long as he's inside that two week window.

I believe you can even choose to Keep this build forever for a couple of your "template" builds, making them more easily accessible.