I'm going to go with the 'imprinting' option. When the dragon hatches, it will imprint on the first character/creature it sees. That may sound really awesome at first, and assuming everything goes well across the intervening decades, it may turn out to actually be awesome for the PC's descendant(s).
Just some of the less obvious issues the PCs will have to face are:
Dragon Fear: Even if it's not actually the game-effect, people don't like dragons. Dragons sow chaos in the lives of mortals, destroying crops, eating herds, decimating populations, etc.
Food Supply: Congratulations! You've got a brand-new, freshly-hatched, baby dragon. Now, what are you going to feed it? That sheep? Sounds good. Now what about tomorrow? What about in a few months, when a sheep doesn't even count as a single meal any more? Are you really going to keep adventuring just to watch your treasure horde dwindle from the costs of keeping that beast fed? An adult dragon may be able to go weeks or months without a huge volume of food, but you've got a ravenous, growing predator on your hands. It wants... no, needs food. "Feed me, Seymour!"
You're "Mommy": Ever tried to get rid of a 5-year-old who really wanted to stick around? Now imagine that delightful child can kill you accidentally, while playing with you. Even if you decide you don't want to keep said hatchling around, you've got to convince it to leave. You might shoo it away, only to have it show up again at the worst possible moment. Or, you leave it staked out in the forest, only to find that it's strong enough to break loose, and meet you in the town market. Or on the roof of the Inn where you're sleeping.
Lodging: I hope you're good with never setting foot in a civilized town or city again. And being forced away from villages by angry mobs who want to kill your new pet. You know how I said they fear dragons? They also hate them. With a passion. As some future paragon of might, you may be able to make the case that your pet can be trusted not to eat everything and everyone in the area. Until you've built that reputation, however, people aren't going to be happy to see your pet. Or you, so long as you keep it. Even if you do manage to keep good PR, your welcome is going to wear thin as you put strain on the local ecology and livestock. Someone's cow goes missing? Someone's child? It's the dragon!