This has happened/is happening to all the sinks in all the houses I've owned. I've also seen it in other people's houses as well (ie, if I'm asked to assist with a home repair). I live in central Texas, in case that differentiates me from other people answering this question.
The only thing I put down these sink drains is liquid soap and water. They have never had any sort of chemical drain cleaners or anything else.
My suspicion is that the metal in these rods is similar to the type of metal used in shower caddies and dishwasher/draining racks. I believe that there is a chrome plating around a much cheaper, weaker metal. Once the chrome has worn off, the underlying metal is exposed and rusts away quickly. I can sometimes find the chrome flaking off of the rod as well.
I tend to solve this problem by replacing the rod. It is overly expensive, yes, and it just begins the cycle anew.
I have tried some other solutions - but they haven't been installed long enough to really say they've stood the test of time. My current experiment is if I can catch one of these early enough (where there is still substantial rod left) then I fill the missing rod chunk with a binary epoxy putty (ie, Waterweld) and then cover it with heat-shrink tubing.
My overall goal is to reinforce the chrome plating with something that will stand up to the rubbing of the sink stopper, and also resist rust/reaction with soap residue.