Most of the dragons in my clan, at least those born into it, get a mix of their parent's names.
Essentially, I take a few letters from each and mash em together until they sound cool. If the parent doesn't have a name, I'll use both grandparent's names, and the known parent's name to make one. (In my lore that's achieved through scrying). If the grandparents are also unnamed that side of the family isn't drawn from, and the name is either purely drawn from the existing parents, a non-related parental figure, or some of the letters are chosen at random.
One of the earliest examples of this is my boi Lamris.
G1's are a different case. They get to choose a name for themselves. Either they can take letters from two dragons they look up to or choose one from any source. Abandoned hatchlings with nameless lineages do this as well.
The only G1 to do this so far is Momoiro, who also happens to be the only G1 that hatched on the cliffs. He would have named himself after his adoptive parents, but after talking with one of the founders he settled on an older Japanese word: Momoiro, meaning 'Pink' or 'Peach-like'.
Dragons can also change their names using the clan's traditions. There are only two who have done this,
Teilr and
Gedam, and the clan has respected this change both by using their new names and allowing the old ones to be forgotten. (Their old name is considered a legitimate source of inspiration for a new one, but those two avoided that entirely)
There's a few that go against the tradition though, since it's not really enforced, just expected.
The three that didn't get names based on their parents are Spruce, Maple, and Oak. Only reason they got these names is because their eggs were stolen by...
something and scattered not to far away, under trees sharing the same name.