This 'Fellowship' comes off the back of a discussion about how else the Fellowship could have been constituted, had slightly different conditions prevailed at the Council of Elrond. Warning: multiple unrelated tweaks to canon incoming. This story should never be written without serious justification.
1) Sam Gamgee. Although Frodo is healed by Elrond of the Morgul wound, he managed to fall off Asfaloth and break his arm in three places. Unable to continue the Quest, he passes the Ring to Sam (with great effort of will).
2) Radagast the Brown. Since Gandalf is still imprisoned in Orthanc (apparently Saruman realised that 'on the roof' isn't the best place to lock him up), Radagast was summoned by Elrond at the last minute.
3) Eowyn of Rohan. Frustrated by Wormtongue's influence in Rohan, she disguised herself as Dernhelm and accompanied Boromir to Rivendell. Sure, the disguise didn't hold very long - but long enough to get her on her way.
(As a result of her testimony, Aragorn and Boromir head straight back south towards Rohan. If Saruman is meddling with Gondor's only ally, that needs immediate attention. The Fellowship won't be going their way)
4) Tauriel of Mirkwood. Totally-not-dead Captain of the Guard, sent by Thranduil to bring word. She has a vested interest/obsession with fighting the Enemy.
5) Thorin Stonehelm. Like Denethor, Dain II Ironfoot sent his son and heir to bring word to Rivendell.
6) Glorfindel of Imladris. While I like the idea [suggested by another participant in the discussion] of Elrond joining the Fellowship, I don't think he'd take Vilya out of Rivendell. Glorfindel is the only elf around (except Galadriel) who can definitely drive off a Ringwraith. We need him.
7) Bard of Dale. Bard son of Brand son of Bain son of the more famous Bard. He chose to accompany his friend Thorin to Rivendell. (Yes, this leaves both realms without their heirs; I think that's plot-relevant)
8) Arwen Evenstar. According to The Laws and Customs among the Eldar, Elves weren't sexist - they viewed men and women as equal, with the sole exception that only women could bear children. This meant that women didn't tend to go to war - and thus, since fighting dulls the elven healing skill (somehow), were generally better healers. 'Indeed in dire straits or desperate defence, the nissi fought valiantly, and there was less difference in strength in speed between elven-men and elven-women that had not borne child that is seen among mortals. On the other hand many elven-men were great healers and skilled in the lore of living bodies, though such men abstained from hunting, and went not to war until the last need.' A psychic healer is a very useful person to have around - and it should be noted that she's probably both capable of and permitted to make lembas (a right and recipe reserved to queens and high ladies of the Eldar). I bet that would come in handy.
9) ... wow, turns out that without four hobbits to make up the numbers, it's really hard to come out with a good nine. I've already got three elves and two humans; there's no wizards on hand, and the only other young dwarf I know of is Gimli. So, since Aragorn's not coming, let's go with...
9) Halbarad the Dúnadan. Kinsman to Aragorn and (in LotR) leader of the Grey Company. He has the skills to bring the Neo-Fellowship through any amount of rough terrain, even when out of his native land. Plus, Don't Panic left me with a soft spot for him.
Obviously, this Fellowship will do things very differently. With no less than four Mirkwood-locals, they would take the High Pass (kept open by the Beornings), and then the Elven Road to Thranduil's halls. After that (and after anyone had left who wanted to - I foresee the entire Fellowship getting caught up in the wars around Dale), they would head down through eastern Rhovanion to come at Mordor from the north. It's possible they would take boats down the Celduin/River Running, maybe even all the way to Dorwinion on the Sea of Rhûn (and the descendents of King Bladorthin?), and actually enter Mordor from the east.
Who would they meet? In rough order: Beornings, Elves, Men of Esgaroth, Dwarves, Easterlings, Orcs. This would be a great chance to explore the Easterlings a bit, and maybe even see the back-lines of Mordor.
And as for the Ring... I'm afraid Eowyn is probably going to try and steal it. Much like Boromir, she has that burning urge to protect her homeland. She may even end up fighting Thorin or Bard over it...
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