With me it's the way a song "sounds" that generally attracts at first. It's not only the melody, but also the timbre of the voice and/or instruments. After the sound, I then consider the words - are they something I relate to, or wouldn't feel out of place singing - A rotund, bearded, middle-aged guy doing a Celine Dion number ?!?

I do think that as one gets older your tastes change. I'm learning and singing/playing songs now that not many years ago I thought were too "old hat", or too over wrought emotionally - for instance I just started singing "Red Wing"(There once was an Indian maid ...)although I've played it instrumentally forever.

I'm also attracted by novelty - just recently, on a whim, I sat down in an evening and learned "Somewhere Over the Rainbow" on the 5-string banjo, with singing. It actually turned out to be easier than I thought.   

Another, that for a long time I felt was too minor/morbid sounding is "Wild Bill Jones" - now, after hearing a local picker's version, I've been playing this single chord wonder a lot.