Elfhelm and Grimbold held similar positions - both were high-ranking Captains of the Mark. Grimbold was a Marshal of Westfold, while Elfhelm was in command of the defense of Edoras after the First Battle of the Fords of Isen. Grimbold was sadly killed in the Battle of the Pelennor Fields, but Elfhelm survived to stand as a witness at Aragorn's coronation, and to be made Marshal of the East-mark - one of the two highest-ranking soldiers in Rohan - by King Eomer.
Grimbold, we know, was a hereditary lord of some kind - his home was called Grimslade ('Hillside dell of Grim'), suggesting that he ruled in the foothills of the White Mountains. He also appears to be a great friend of Elfhelm's - and since Elfhelm is connected to Edoras, that suggests that Grimslade is probable in the eastern parts of Westfold, near to the capital.
But what about Elfhelm? The '-helm' of his name is clear - it means helmet - but is 'elf' really 'elf'? There are four names in Middle-earth recorded with 'elf' at the beginning: Elfhelm, Elfhild (wife of Theoden), Elfstan (grandson of Sam Gamgee), and Elfwine (son of Eomer). Of these, the latter two are actually Old English translations of pretty famous names - Elfstan means 'Elf-stone', or Elessar, while Elfwine means 'Elf-friend' - Elendil.
So we can assume that Elfhelm is, indeed, named 'Elf-helm', or maybe 'Elf-protector' - and likewise that Elfhild, who died in childbirth, has a name meaning something like 'Elf-battle' (by the literal meaning of 'hild'), or 'Elf-fighter'.
Those are very unusual names in Rohan. There are no other names remotely like them in recorded Rohirric history, and they even have a common theme besides the 'elf' - both are connected to war. A mysterious, high-ranking soldier who is very close to both the son and nephew of the king, who shares a naming theme with the king's sister? I think you'd be hard-pressed not to assume they were relatives.
Elfhelm may well be the king's brother-in-law. Theoden was thirty years old when Theodred was born and Elfhild died, but that doesn't mean she was the same age. In fact, the only hint we have suggests she may have been younger: Morwen, wife of Thengel, was only 21 when they wed (though he was 38!). There is no indication of precisely when Theoden and Elfhild married, but in a world without reliable contraceptives, first children are often born pretty quickly. So Elfhild may well have been 25 or younger when she died.
If Elfhild was the elder sibling, and her parents followed the pattern of Rohirric and Gondorian nobility in spacing their children some five years apart, Elfhelm could have been 20 (or even younger) in 2978, at the birth of Prince Theodred. That would still make him some sixty years of age in the War of the Ring - which would fit his high-ranking position, even if it seems a little old. Instead of a brother, he could be a nephew (in a whole family of elf-centric names) - or simply an even younger brother (Theodwen was younger than Theoden by fifteen years - and had another sister older than him!).
And so, in his position as kinsman (but not in line for the throne), Elfhelm would have been given command of the Muster of Edoras - and responsibility for training both Theodred and Eomer in battle. And, when he discovered that young Eowyn was just as fervent as her brother about fighting the orcs who had killed her parents - and remembering that his own parents had given him the name invoking armour, while his sister had the proper fighting name - maybe he took her under his wing as well - teaching her the skills that would one day let a twenty-four-year-old woman of Middle-earth bring low the Witch-King of Angmar.