Someone said "if", and my ears pricked up...
The premise is that Eomer is banished from Rohan, and doesn't run into Aragorn et al on his way out. Perhaps Boromir tripped and fell off the Bridge of Khazad-Dum, so the entire Fellowship has headed east. Doesn't matter - all that matters is that they're not in Rohan.
If we're going to send him to meet the Elves, he's going to need a reason - and 'shelter and protection' doesn't cut it. Eomer isn't the type to run scared, so if he's leaving Rohan at all, it's either because a) he feels honour-bound to obey even an unjust command, or b) he has a plan to take the country back.
With that in mind, let's see if we can take him to see the Elves?
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Mirkwood: The easiest way to get Eomer into proximity with Mirkwood is to send him up Anduin to the old Éothéod. He's utterly convinced that he's failed his king, so he's straight-up leaving it all behind. He heads all the way to northern Mirkwood to reclaim the ancient capital of
Framsburg and found a new kingdom.
If we assume the West doesn't fall in this scenario, and the Ring is still destroyed, then Eomer's new kingdom will be perfectly positioned to tap into the Mirkwood-Esgaroth-Dale trade route come the Fourth Age. All of those settlements were afflicted by the war, so a new, unbattered economy (especially one that may be able to supply dragon gold from the cold-wyrms of the Withered Heath, a la Fram) is going to do well. So he'll have a nice trade relationship with Thranduil.
He probably feels a bit guilty about the way Rohan was destroyed by Uruks, though. And I doubt War-Queen Eowyn will ever forgive him.
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Rivendell: This is the tricky one. How do we get Eomer to the House of Elrond? Well... let's send him earlier! While Boromir is riding through Rohan on his mission to take Faramir's dream to Imladris, his horse trips on a rock. Poor Boromir breaks both his legs, and Eomer (who was on rock-clearing duty that week) feels honour-bound to take up his quest on his behalf. Off he goes, and winds up at the Council of Elrond.
Let's say he joins the Fellowship, but ducks out before they enter Moria; we know he's kind of spooked by haunted caves. He probably justifies it as taking word to Theoden and Denethor. Unfortunately, by the time he arrives, Theoden is already well under Saruman's influence, and kicks him out.
Would he go all the way back to Rivendell? I think maybe -
if he knew Saruman was involved. Taking word to the Wise that Saruman was in full Evil Overlord mode might have seemed like a good idea, especially if the Riders now believe Eomer abandoned them and so won't follow him anyway. Elrond, needless to say, would be happy to take him in, but wouldn't really do anything to help.
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Lorien: Far and away the easiest. "If the Wizard of the Ironstone Tower has ensorcelled our uncle," says Eowyn as Eomer prepares to leave, "then might not the Witch of Dwimordene restore him to his mind?"
Would Galadriel help? I think she'd be up for feeding and watering Eomer's band of adventurers, but she's not one to offer direct military aid. More likely she'd give him some words of wisdom, like "The voice of Saruman is a WORM that the TONGUES of Men should not repeat", in the hopes that he'd get the hint and go throw Grima off a cliff.
hS