Use the word scion when talking about a young member of a family that is known to be wealthy, powerful or otherwise important, such as a prince, heiress or the children of, say, the President.
Scion sounds a little bit like son, which is helpful because it almost always means the son, daughter or descendant of a rich or prominent family. Its earliest examples were used to refer to the young shoots of larger, older plants. It's not surprising, then, that over the centuries its meaning has shifted to include the human offspring of certain well-established families. These days it's invariably used to talk about a person such as Prince William or, say, the late John F. Kennedy, Jr. — both of whom are or were scions of their respective families.