Second gen characters are not officially counted when it comes to activity checks, so sometimes a dropped character can go without being noticed and make the place look untidy. At some point there may be a (much looser) second gen activity system in place, but until that time there is this.
If you drop a character, or lose a character to an activity check, you are free to continue to play that character's child only until somebody else takes them, at which point (or sooner) you ought to drop them - please, if you do so, post about it here so we can tidy up the accounts and strike out the bio, so people know who's still active and who isn't.
If you're playing a child and the other parent leaves, you may continue to play that child and they just become a sort of AU possibility - their relationship with any other child of that character is to be negotiated between yourselves.
Otherwise, if you simply want to drop a second gener, again, please just let us know here. Second gen characters are ONLY adoptable by a player of either parent, and only with the express permission of the original player.