有什么好 + [verb / adjective] + 的
没什么好 + [verb / adjective] + 的
有什么好 + verb + 的 asks "what's there to do that?" and means there is nothing worth doing it about. The verb is usually a reaction – worrying, being afraid, crying, looking, talking about.
An adjective can go in the same slot: 有什么好奇怪的 is "what's so strange about that?"
没什么好…的 says the same thing as a flat statement: "there's nothing to ...".
With nothing after 的, the question is rhetorical. With a noun after 的, it is a real question about what is available.
The final 的 closes the frame. Without it the sentence is unfinished.