Some of us (myself included) don’t really have a problem with “eonian” referring to that which is truly eternal, namely God, when it refers to God.
“Eonian” does not refer to God here, of course, but to the resurrection body and/or the “transcendently transcendent burden of glory”; but some of us (myself included) think the term can describe things less than God as having a lesser, derivative eternality. Since such a lesser eternality could be revoked by God (such as at the end of Romans where the eonian times of the secret have ended thanks to the eonian God), or might only be described as “eonian” as a figure of speech, the question of whether something is never-endingly eonian or not would have to be settled by context, not by appeal to the term.
Alternately, some of us (myself included) tend to think that “eonian” in the New Testament was used as an adjective to describe the quality of something being uniquely from God. The eonian God (as at the end of Romans) being uniquely from God would refer to the Son; and the glory and the resurrection body (which is not yet seen but hoped for) which Paul is speaking of here are both uniquely from the heart of God and nowhere else. This term usage is also neutral to the question of whether something other than the everlasting God goes on permanently after being established by God: eonian life (which is the life of the resurrection Paul has in view, not whatever life the impenitent wicked have at resurrection) continues forever by God’s grace, and also happens to be the life of God Himself, always existing, never beginning and never ending, shared with creatures; whereas eonian chastisement (as in the distinction of Matt 25 between the sheep and the goats) may or may not have an end, but still comes uniquely from God (as the highest and final authority). Either way what is eonian comes uniquely from God alone, and either way we have to figure out from context whether the object of the adjective is supposed to continue never-endingly.
I know some universalists go with the idea that “eonian” means, not “uniquely pertaining to God” but “pertaining to the age to come”. Eonian punishment or eonian fire is the punishment or fire of the age to come, eonian life is the life of the age to come, the eonian God is the God of the age to come, our coming resurrection and/or our transcendent glory is the resurrection and the glory of the age to come. That age may be the never-ending final Day of the Lord, or a millennial reign sub-age (which is itself part of the final Day-Age); if the latter, then (according to proponents of this notion of “eonian”) “the eonian God” refers to the role the Son takes while reigning before and until putting all enemies under His feet so that, having finally submitted all enemies to Himself He gives up the kingdom in submission to the Father. His reign might or might not continue in other ways (depending on the Christology of the proponent of this meaning of the term), but the eonian life of the faithful which comes into its glory during that reign continues on after this sub-age into the all the ages of ages to come in the final Day.
I will point out that an ECT proponent or an annihilationist could accept such a definition of “eonian”, too; they would be deprived of one linguistic evidence in favor of their positions, and a particular ECT or anni might (mistakenly) believe their position depended totally on the linguistic evidence of the term always meaning nothing other than never-ending, but educated ECTs and annis know their position is based (rightly or wrongly) on more than that meaning of the term. The alternate meaning proposals (and some kind of alternate meaning has to be true because there are several times the term is used to describe something that does demonstrably have an end) are neutral to the question and so don’t really count for or against ECT, Anni or Kath.
What’s most important about the term from our apologetic perspective is that it doesn’t count specially for ECT or Anni, and not specially against Kath.
I think that succinctly covers the various options and issues. Relatively succinctly. 