What is Doon's favorite bug?

The story doesn't specifically say what Doon's favorite bug is. However, a close reading of the text does tell us that Doon has "always been fascinated by bugs." If we were to speculate, Doon's favorite bug may possibly be the moth.
In the story, we learn that Doon keeps a bug diary called Crawling and Flying Things. His personal notebook is filled with drawings and facts about various types of bugs. Since the book has a general title about "crawling, flying things," Doon has included beetles, bees, spiders, and moths in it. Spiders aren't insects, however. They are arachnids.
In the Pipeworks, Doon finds three interesting bugs. One is a black beetle and another is a moth. The third, however, has an interesting spiral-patterned shell on its back. Doon is fascinated by this bug but ends up killing it accidentally.
For a time, Doon keeps a cabbage worm in a wooden box. In due time, the worm turns into a beautiful light-brown moth. Doon is completely fascinated by the metamorphosis process that turns a worm into a stunning creature with light, fluttery wings. As he watches the moth fly away, he has an epiphany:

Doon watched until the moth disappeared. He knew he had seen something marvelous. What was the power that turned the worm into a moth? It was greater than any power the Builders had had, he was sure of that. The power that ran the city of Ember was feeble by comparison—and about to run out.

So, Doon's favorite bug may well be the moth.