The problem is, then you lose the damage +1 bonus, so its only a net damage +1 gain, and that’s for discarding the axe.

As its the flat-out damage bonus that makes the card overpowered (see now obsolete Rider’s Spear, Rohirrim Axe, Sword of the Mark, etc), my suggestion is to drop the damage bonus, or make the damage bonus conditional, like all the other weapons. Flat-out damage makes the card OP, not the ability. So I would either make it conditional or drop damage altogether.

Rohirrim Sword
Possession • Hand Weapon
Strength +2
Bearer must be a Man.
While at a battleground or plains site, bearer is damage +1.
“...long swords were at their belts, their burnished skirts of mail hung down upon their knees.”
generally damage+ isn’t all that useful, but it keeps with the theme.

Rider’s Axe
Possession • Hand Weapon
Strength +2
Bearer must be a Man. Bearer is damage +1.
Skirmish: If bearer is mounted, discard this possession to make bearer strength +3 and damage +2 until the end of the current skirmish.
“‘The men of Rohan must ride forth today, and we will ride with them, axe, sword, and bow.’”

Okay, NB. Let’s say I change the damage to conditional and lower the cost to the more normal . What would be a good conditional way to have that damage become active, in your opinion?

I don’t play rohan decks, and forgot about swordthain.

YOU DON’T?!?!?!

QUICK! Grab him Eowyn, Lady of Ithilien, 4 Riddermark Javelin, 4 Sword Rack, Coif, Rohirrim Mount, Sworthain, 4 Rider’s Mount and 4 Rohirrim Helm!

That configuration will make a deck to be reckoned with, whatever else you may add in.

Horse of the Mark
Possession • Mount
Strength +1
To play, spot a Man. Bearer must be a hunter companion.
At the start of each skirmish involving bearer, each minion skirmishing bearer must exert.
“...he saw not a shadow, nor a blur, but the small figures of horsemen, many horsemen, and the glint of morning on the tips of their spears was like the twinkle of minute stars beyond the edge of mortal sight.”

Not convinced DI. Not sure the point but i dont think it needs to exist.

Éored Steed
Possession • Mount
Bearer must be a Man.
Each time bearer wins a skirmish, you may exert a minion (or wound a minion if bearer is a hunter).
“Merry and Pippin heard, clear in the cold air, the neighing of war-horses, and the sudden singing of many men.”

Cool take on the horse thing.

Bow of the Mark
Possession • Ranged Weapon
Bearer must be a Man.
If bearer is at a battleground or plains site, he or she gains hunter 1.
At the end of each archery phase, if the fellowship archery total is 1 or less, make the fellowship archery total +1.
“...some of the horsemen had bows in hand, and their arrows were already fitted to the string.”

Hmm ok. Seems like it needs a tweak but not sure what.