Éomer's ability could well be the ultimate form of action advantage - without using any other cards to make this happen, he quests, he attacks, and, assuming he takes his target out, he frees up another character from having to defend against the now-destroyed enemy's attack. Destroying an enemy also removes its threat from the staging area, which means Éomer contributes more than his willpower to the questing. And it's just cool to get the first strike in. And he does this at the fairly reasonable cost of one resource per use. But you need time to set that up, and you need to be playing the right deck and/or the right scenario. If you just put him in any old deck, he won't have any targets to use his ability on, because any enemies you revealed last turn have already engaged you, and if you play him in the wrong scenario he won't be able to kill his targets. My limited experience shows that Éomer is good out of the gate in most Khazad-dum/Dwarrowdelf scenarios, since they often have goblins that pop up from shadow effects and don't get to engage you before Éomer has a shot at them - and they're frail enough for him to take them out by himself.
But if you're playing against something other than goblins, Eomer will need some help. That help will generally not be coming from , so I'd advise against playing him in a mono- deck. Here are the main things to consider:
1) You need to ensure there are targets for him. I can see three ways of doing that:
- Change the timing of when Éomer strikes. The main tool for this is Late Adventurer - don't commit Éomer if there isn't already an enemy in staging, and wait to see if one pops up. In theory you could also give Éomer the icon, give him Windfola and use Don't Be Hasty! to remove and recommit Éomer when an enemy pops up. That's entirely too many combo parts to put together with any kind of reliability, but if done it could allow Éomer to strike three times in one turn - commit and attack as usual, Don't Be Hasty + Windfola when an enemy pops up and then Late Adventurer at the end of staging (which you can still do because Windfola exhausts itself but not the attached character).
- Keep your threat low so the enemies don't engage you on their own. Fastred seems to be the easiest way of doing that - particularly because if any enemies do get through, he can bounce them back to staging again. In-sphere, Frodo Baggins could be helpful, but having two heroes spend their regular income on their own abilities every turn sounds like something that would slow a deck down quite a bit.
- Keep enemies in staging with card effects. I can see little synergy with traps - Ranger Spikes is the one that keeps enemies in staging, and because it also brings the trapped enemy's threat down you don't really care about destroying it (unless the trap attaches to the wrong kind of enemy, like a Goblin Sniper) - but if you're playing opposite a deck they could be bringing ways to increase enemy engagement costs or cancel engagement checks altogether. In-sphere, Terrible to Behold will help you deal with an enemy that did get to engage you - and, conveniently enough, Éomer is a Noble and can use it if there was nothing there for him to attack this turn.
2) You need to ensure Éomer can actually kill something, or else his attacks are meaningless. The most obvious and effective attack boost is probably Spear of the Mark, and it's what I've been using, but once I get that pack I will also be trying Firefoot; that's more expensive and the response won't be useful if Éomer mops everything up in staging, but sometimes an enemy will still slip through and be there in your face (unless you'e using Fastred, I guess). The other way, if you don't want to include in your deck, would be to make Éomer an Elf-friend, and give him Light of Valinor and a Mirkwood Long-knife or two; it's another thing I'm theorycrafting on because I don't have Sands of Harad, but it would make more use of Éomer as a quester (his normal 1 is a bit sad otherwise) and let you get double use of the attack boost.