Has any narrative attempt ever been made in DC comics to link the Emerald Eye of Ekron’s power with that of the Green Lantern Corps’ rings and power batteries?
It seems like such an obvious call I’d be surprised if the answer is “no.” But I’ve been a loyal reader of Legion of Super-Heroes for decades, and I don’t think I’ve ever seen such linkage made.
I feel certain cmkeller will know the answer to this.
Yes. Just last month, in fact. 
Care to go into some detail? I’m happy to be spoiled.
The Emerald Eye of Ekron belonged to the Green Lantern who was in charge of Vengar’s sector until it was recently destroyed. It seems to be his (Ekron’s) equivalent of the ring, since it holds the power, and taking it out of the Emerald Head of Ekron (Ekron’s ship, by all appearances), seems to have taken away his GL abilities.
If it were just a plain old “power ring,” then it would need a battery.