Very neat topic, thank you for posting.
What first comes to mind for me is my 1953 Bowman Bobby Shantz card. Shantz, who pitched for the Philadelphia A's and later the Yankees, was the '52 A.L. MVP when he won I believe 24 games. Anyhow, though this card is regarded as a common to most and is not particularly valuable or difficult to find - it has a lot of sentimental value to me. Shantz was my Dad's favorite player growing up - he lived near the Bridgeport / King of Prussia area in PA - north of Philly. When I first got into cards as a kid in the late 1980's, I naturally asked Dad where his baseball cards were - no luck there, they had all been thrown away. (Yeah, thanks Nana!) But when I asked who his favorite player had been, his eyes lit up and he said "Little Bobby Shantz!" Dad never really cared much about sports as an adult - I mean he would watch games, but was never any type of super-fan. He certainly never cared much about baseball cards, but he humored me as a kid. I got him the '53 Bowman Shantz for a Father's Day present from a local shop one year when I was probably 12. We lost Dad 10 years ago this past summer, and having that card now is always a memory of him and thoughts of the games he must have watched at the old Shibe Park as a kid. It's only in about VG condition, but priceless to me - now at age 41.
Other than that, I did save a few vintage cards that I had traded for as a kid - not in the best shape, but sentimental to me due to those memories. Yes, dealers used to trade with kids! The best are probably my '56 Mantle (now graded SGC A) and Ted Williams (SGC 3).
You don't have to explain about something seemingly worthless being priceless in your eyes. This past summer I spent probably several hours - and finally was successful in finding - a box of cards that had a toploader in it with a price tag sticker on it from a long gone, but favorite old baseball card shop I used to go to in Charlotte, NC with my mom - in fact, the same one we got the Shantz card at. Yes, I was looking only for an old price tag. It meant something to me because it was a memory of such good times.
Remember the movie "Throw Momma From The Train" with Billy Crystal and Danny Devito - the scene where they are examining Owen's coin collection - and Billy Crystal realizes it's not about the coins at all - it's just change that Owen got years ago as a kid whenever he was somewhere with his Dad. A simple grasp for the past, and a tie to something that you can no longer have. I get it. And yes that can totally happen w/ baseball cards.
-John