1. Check all of the shift cable housing ends. When I have encountered a derailleur that would drift out of adjustment that quickly, what I eventually found was a bad cable housing end that was gradually shortening itself through use and screwing up the indexing. You could adjust it, but the cable housing would just shorten itself a little more and drift out of adjustment again.
2. In order to index properly, the derailleur has to move in the same plane as the cassette cogs. It's common for a bike that's used daily to bend the derailleur or hanger out of alignment. Assuming this is a claw mount derailleur, shift into a gear that makes the derailleur arm point straight down. Then look it from the back to see if the derailleur arm seems to be pointing toward the tire. If it is, just bend it back with your hands. It's pretty easy to get a 7-speed in alignment well enough to function. 9-speed and up I prefer to use an alignment tool.
3. Regardless of how you measure it, I'm betting that chain is worn out. If that's the case, I'd give it a test ride with a new chain but my bet is you are going to need a new freewheel too.
4. If you can feel slop in the derailleur pivots that's probably worn out too. If I were doing it, I'd probably just replace it with what was there previously.
That's quite a few parts and we haven't even talked about brakes yet. Department store bikes are so cheap to buy that the economics of tuning one for constant reliable use tend not to pan out very well. The parts bill for the stuff indicated above will come pretty close to the original cost new. What you'll end up with, however, is a crummy old bike that has a few new replacement parts. A new bike will have every single part brand new, every part designed to work with every other part, and come with a new bike warranty.
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My greatest fear is all of my kids standing around my coffin and talking about "how sensible" dad was.