A summoner can invest a weapon to share its benefits with the eidolon.
There doesn't seem to be any restriction on which weapon is invested.
So a summoner can invest a short bow with a striking rune, and the eidolon gets the benefit on its strikes.
The rule seems to explicilty allow the eidolon to gain the benefit of the runes even though its using a natural unarmed attack and not a weapon.
Certain runes are limited to ranged or melee. Or to particular types of weapons.
The way I read it the eidolon can get around these restrictions totally. Note that the eidolon can get a natural ranged attack. So they could get a ghost touched ranged attack?!
Not especially powerful. Just interesting.