Normally, I'm no fan of using an original character as the central part of the story. It's typically even worse to introduce an original character be the love interest of a terminally single character. The author of two related stories breaks both of these rules, but she does it so well that I add this into my own personal canon.
The Girl Who Aimed Straight is the story of one Lily Withypoll, who was given an elvish bow and arrows by one Bilbo Baggins. Gandalf gives her the challenge of forming a school, and of teaching hobbits their history, and her story is woven in through Frodo, Sam, Merry, and especially Pippin's histories. It's an absolutely beautiful story, and its sequel
The Jewel of Brandy Hall is just as worth the time. (Fernwithy, the author, also has fantastic Harry Potter stories that I'll mention in a later post. "The Doll Army" is an all-time favorite in the fandom.)
Marbles and Slingshots fits neatly into the background of the canon story, in the years between the death of Frodo's parents and his adoption by Bilbo. I've read nearly everything that author has posted, as well, but Marbles is still one of my favorite stories.
If you want dark!fic, edging quite handily toward the darker edges of that genre,
The Captain and the King does what most fics that mention torture try to do. As you can probably guess from the title, it involves Boromir and Aragorn. Saruman is also a main character of the story, which just might say all of the rest.
All of the above are sad/bittersweet at best and utterly dark at worse, so I'll add on some fluff to make the list a little better.
Quarantined is another alternate past, but by a different author. It's an alternate bit of history, but nothing all that unusual. When Aragorn takes ill, and Gandalf needs a suitable place to keep a sick Man, the wizard remembers that Bilbo keeps man-sized rooms and has already had the quite-contagious disease. Despite the setup, it's quite fluffy, and involves a young Frodo befriending Aragorn.
Yada Yada Yada, by the same author as Marbles, is the most efficient summation of Elrond's Fellowship of the Ring council you could hope to find, and
Nine Men And A Little Lady is a classic of the fandom (and a warning against OCs that aren't done absolutely on target.)