Alia, that's a novel approach!
Here are the results of my experiments:
The plumbers tape didn't work great over time, because I need to be able to take the handle off when I am not working on poly clay (my pasta machine is mounted on my desk just to the left of my keyboard pull-out shelf - and since my day job is architecture, I have to be able to convert from one specialty to the other!). If you can just leave the handle in the machine, I think the plumbers tape is the way to go, but it gets ripped up when you push/pull the handle in and out.
I almost ordered magnets from the site that Rhoda posted, but, although they are only $0.20 each, the minimum postage charge was about $7. Instead, I found a package of 6 tiny (1/4" x 1/16") magnets for $5 at Hobby Lobby here in town. I placed one of them on the end of the handle. Although it took a bit of time to get the magnet INTO the pasta machine hole (it wanted to stick to the outside!), I was finally able to do it. The magnet is now stuck deep in the hole where the handle gets inserted. I can still remove the handle as needed but the magnet seems to be strong enough to keep it from falling out.
I might now order a new handle (if that's possible) since my old one is broken from crashing to the floor so many times. It's still functional but it can pinch my hand if I am not careful.
Thanks, all, for your suggestions! I hope this post helps someone else!!