Éomer Éadig at EoA
There's also a little backstory on influences here:
Éomer Éadig, ANCESTOR OF SIR WINSTON???!
Wiki has little to add to EoA on the fate of Éomer himself, save that Elfwines mother was Prince Imrahils daughter Lothíriel, and that "Éomer's sword was called Gúthwinë (Old English battle-friend)." He was 90-91 at his death.
His son Elfwinë succeeded him.
Tis also mentioned in the Appendices to LOTR that "often the hooves of the Rohirrim steeds were heard on the fields of Rhûn and Harad" (or something in the like), which means that during the first years of the Fourth Age, he and Aragorn were mostly busy mopping up the lands which used to be loyal to Sauron.