I'm going to be running Tyranny of Dragons soon and I allowed my players to come up with their characters before the session zero with the caveat that their characters may need to be tweaked based on what we decide in session zero. Two of them want to play genasi who are searching for their genie parent, among other goals. One of them wants it to be part of the inheritor background. I'm struggling to come up with ideas on how to provide emotional payout for this idea, particularly since the genie rarely cares for the offspring. I do have ideas on how to actually find the parents. Any suggestions? I would like to have options before the session zero so I know if we need to go back to the drawing board on these backstories based on what my players tell me they're looking for from this idea.

Depending on the tone of your game, it could be that the genie truly doesn't care about or even remember his children. Alternatively, the genie could have heard a prophesy long ago, something like the most worthy of his offspring can stop Tiamat's entry into the world if they take his ancestral scimitars (or pair of other magic items). Unfortunately for him, the genie got around, and has hundreds of genasi children by different mothers, so he doesn't know which to give the scimitars to. So, to determine the "most worthy" of his children, he hid himself away and waited to see who was skilled, caring, and perseverant enough to hunt him down. When the characters find him, he gives them the scimitars and tells them to go kick the dragon cult's butt. (If you actually hand out magic items, though, be sure to have the genie bestow equally cool if less prophecy-laden items to the rest of the party for helping his children through all those dangers.) And of course, always remember that the genasi might die, so be careful not to send the campaign too far away from the main plot! Good luck and have fun.

Inheritor Background -  you have a newly acquired tattoo imprinted on the inside of your sword arm extending from your wrist to your elbow; and have no memory of how it got there, or where the tattoo came from. (This could also be done as a change or sudden appearance of their Szuldar lines for a genasi).

They eventually find out that the tattoo (or szuldar lines) are actually clues to undoing the 7 locks holding their parent prisoner. Every time they solve the clue and then succeed in opening a lock, they get a sending from their parent and then the tattoo changes to the next clue. The sending they get is the only way they can communicate with their genie parent. You can build the relationship through the 25 word sendings the genie sends them when they succeed in opening a lock. It can start cold and demanding and graduate to warm and proud over time.  Being freed by their genasi offspring earns the genie’s respect and cause the genie to actually care about all their genasi children and to reward the PC (now their favourite child) with a single wish and a permanent seat at the table at the parent’s right hand side.