I suppose the child gets a claim to any personal effects of the parent. If the parents role in society offers any boons not available to an ordinary citizen maybe those too.
But in this situation it would probably just be a free shot at joining Starfleet.
There isn't, except when there is. And when there is, they're called credits, but you can't spend them in the Federation, except (I guess) at starbases like Space Station K7, so it's not entirely clear why other governments consider them to have value.
I just wanted you to know that I save these for later, and having the captions be several seconds into the gif is part of the humor. But damn does it make it hard to find/remember which ones have which burn in them.