Why is Legolas’s Character so Underdeveloped?
In a lot of ways, that’s exactly what happened with Legolas. In the earliest drafts of The Fellowship of the Ring, the elvish member of the fellowship isn’t Legolas, but Glorfindel. It wasn’t until much later that Tolkien changed his mind and replaced Glorfindel with a brand new character: Legolas.
So, part of the lack of information about Legolas is simply just that he was a newer character, so Tolkien didn’t spend the same time developing his history as he did with some of the “original” characters.
Beyond that, though, the very reason why Legolas was created explains why we learn so little about him. Simply put, Tolkien made Glorfindel too powerful, and so he needed to replace him in the fellowship with another elf that wouldn’t overshadow Aragorn. ((EDIT: A major theme of LOTR is the ending of the time of the elves and be beginning of the age of men - so it’s important to the story that the elf in the group doesn’t hog the show. And both movie adaptations of the books have emphasized this problem with Glorfindel by cutting him out of the story entirely - Bakshi’s 1978 animated adaptation replaced Glorfindel with Legolas, while Peter Jackson’s live-action adaptation replaced him with Arwen. In both cases it served to shrink the cast and replace the super-powerful Glorfindel with a less-powerful elf.)) Since Legolas’s primary objective, from Tolkien’s point of view, is to represent the elves while still allowing Aragorn to shine as a leader, it’s not surprising that Tolkien doesn’t spend too much time developing Legolas’s character and backstory.
I think (and this is just my opinion) that Tolkien tried to “cheat” on Legolas by making him Thranduil’s son - thinking that, since we already know Thranduil and the elves of Mirkwood from The Hobbit, that this was a sort of “instant backstory” - in a lot of ways similar to how a lot of the information we have on Gimli is based on information we already had about Gloin from The Hobbit ((EDIT: Plus, the rocky relationship between their fathers makes for the great side-story of Legolas and Gimli’s friendship.)) But we still have more information on Gimli than Legolas (and the difference seems more extreme since we’re used to the dwarves being “mysterious” and getting little information on them anyway, while usually you can’t get Tolkien to shut up about the elves.)
Plus, it’s just awkward that Legolas doesn’t appear in The Hobbit. (Say what you will about Peter Jackson including him in the movies. But I’ll bet you what little money I have sitting in my bank account right now that, had Tolkien had the opportunity to re-write The Hobbit after publishing LOTR, he would have included Legolas somehow too.)
Anyway, to wrap this up: It’s not so much that Tolkien forgot about Legolas. It’s just that, he was a last-minute addition to the cast, and it was Legolas’s job to basically stand there and look pretty while Aragorn and Frodo saved the world. Not too much backstory needed for that. *shrug*
SOURCES: LOTR, The History of Middle Earth vol. 9