Yesterday, I grabbed a monitor that had been put out for trash. It was an old 1702 monitor and it was covered with a semi-clear plastic dust cover. This cover was dry, but it felt sticky. Not super sticky, like you knew something had been spilled on it, just semi-sticky, like it makes your skin crawl to touch it.

This isn't the only time I've seen this happen either. A woman I know had dust covers over her Dell system and monitor and a year or so after her death, I went to power it up for her roommate and all the covers felt sticky.

She did smoke, however nothing else in the house is even discolored from it. None of the plastic bags in the house feel sticky. It's just the dust covers that get like that.

Do they make them out of some kind of special plastic that attracts nicotine? Or is there some other reason that they turn sticky?

Was the only one thinking of the obvious answer to this question???

Or do I just have a sick mind???
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Yes!

In earnest: soft, flexible plastics are made using plasticizers, giving them their flexibility.
Some plasticizers, when they evaporate from the plastic, may cause it to become brittle.
Others may deteriorate over time and cause the nasty side effect of stickyness.

more on plasticizers can be found here

I have 2 Commodore vinyl covers for VIC-20/64 and another one from the Amiga era and none of them are sticky.

The Commodore covers date back from 1981/1983 and the Amiga keyboard cover was made in 1992.
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The couple of original covers I own are in good condition as well. Quality and mileage may vary, I guess.