Hi John,
I always carry a folded up piece of paper and a pen, or if I'm at home have a notebook handy, in case inspiration hits - now I also sing it straight into a phone to make sure I get the tune as well. Before smartphones I'd note down the song or songs it sounded a bit similar to.
The thing about having ideas for songs is that they often come at inopportune moments, so you've got to have something handy to quickly note it down, no matter what you're doing at the time.
I type the song up and chords on the computer and save it - all songs saved under the year I wrote them. Most never see the light of day again, but just once in a while there's be one I'll feel is worth printing out and then I sing and play it over and over, scrawling changes with a pen on the printed out sheet until it's so illegible that I print out another copy with the amendments typed in, and go through the process again until I feel it's ok - or until the nagging doubt about certain lines goes away.
I have a Zoom HD recorder that acts as a mic and can connect that to the computer (which is usually on) so I can make a quick demo for myself too using Audacity at each stage, deleting the previous version when I record a new one. I save these by year and month..
I've tried idea generators and fed what I've thought were interesting song titles to ChatGPT and it's come up with the most lame and cliched rubbish you can imagine, complete with forced rhymes and awkward phraseology. I've used Rhymezone occasionally but never successfully - usually if I'm down to that then the song's dead and the fragment I've written goes into the ideas folder.
I always try to make a note of any and every idea - it's easy to push ideas aside because "that's not a proper song idea" or "it doesn't sound like what I'm aiming at" - but they can sometimes be the most rewarding ones - not least through being unexpected.