Character Side Plots are a staple of RPGs, and other forms of RPGs. They can make some very cool moments that reveal a lot about the character, and they can provide a good break from whatever major plot you're running - without actually breaking tension. Oh, and as a general rule of thumb, your players definitely want them. Very few players don't want a side story for their character. Maybe they don't want to be center stage and in the limelight - because that is scary - but they want a story for their character that's more than just "that person who was with the group that did the thing."
With that in mind, let's talk about character side plots.
Hinting and Build Up
Some of the best side stories are the ones that are hinted at for a bit first. It's kind of like an appetizer. You get a hint that this character is more than what you've seen already. That there is more going on than what you may have assumed. Perhaps it's just knowledge you don't expect them to have. Perhaps it's a person from their past showing up.
The idea here is to take a queue from serial television. You don't just dump the whole thing in one go. You hint. Then you let it go for a while. Then it comes back. Eventually you answer the question and with that everyone gets to see more of what is going on, and perhaps a bit of closure and character development.
Remember The Other Players
You know what sucks? Sitting at the table for one, two, or more sessions twiddling your thumbs with nothing to do while the GM runs a plot with one other character - or a couple characters - on a side story. It's actually a great way to lose players, because you go a couple sessions with nothing to do and it gets into your head that you're just sitting around watching everyone else have fun at the table.
Don't let this happen when you are GMing. Yes, the side plot is for the specific character, but have things for the other players to do as well. And yes, I mean every session. Who the PC travels with- who they are now - should be important to the story about who they were. How do the other PCs help them out with this plot? How do they help them deal with the aftermath? The focus should land naturally on the target PC, but no one should be out of the circle.
Take Care With Backstories
Players - some players - will seed their backstory with hundreds of hooks for potential stories. Others will leave their back story very simple and plain. Either way, be careful before playing with the backstory. Unless you have worked out with the players what is sacrosanct and what you can mess around with, don't go smashing houses without talking to the player in question. You don't have to give any spoilers or anything - though you can if you want to have the players help going into things - but a simple heads up of "Hey, I want to bring something up from your past, is that ok?" can go a long ways.
Don't Kill Family
Honestly, if you can avoid it, don't kill a PCs' parents/family/hometown if you can avoid it. It's basically a trope at this point. Killing the family/friends/town in the backstory is done to get emotional investment, but it's honestly kind of a cheap move. If you can't get around it, go for it, but try to make it special. If you're thinking about doing it, try to find a different way. Everyone will thank you for it.
You Can Work With The Player
One of my favorite ways to do a reveal of a character, or a side-story thing, is to work with the player for how things may turn out. The idea is you partner with the player to make an experience for the other characters. The other PCs are part of who the target PC is now, and so you can use that. You and the player flesh out what can happen, and work to bring about certain things. The other PCs then get to decide how it plays out with their own choices.
I love this because it does all you want out of a PC side story, but you also keep the other players in mind and put it on kind of as theater for them. It's worth trying.