As far as I can tell, going down through the list of the uses of eonian in the NT, there is only one time in the NT (but there might be others in the Greek OT) when the phrase “theos aio_nios” occurs, and that’s Rom 16:26: “from (or perhaps in accord with) the injunction of the eonian God being made known into all nations, into faith-obedience.”
Would it be redundant to translate this “in accord with the injunction of the God from-God”? (or “of the God-essential God” or “of the God-uniquely God”?) Only to someone who denies the divinity of Jesus! In that case, yes, the translation would be redundant, and some other translation would be more appropriate. If Jesus is somehow “very God of very God”, though (even in some non-trinitarian ways), then no it isn’t redundant. It might even correspond to “the heralding of Christ Jesus from the revelation of a mystery hushed in times eonian, yet manifested now and through prophetic scriptures” (which was the previous verse in Rom 16.)
That having been said, I should correct myself about the reference of {aio_nios} in Rom 16:25, since it is to “times” not to the “secret”. The doxology there reads (rather confusingly) in Greek: “kata apokalupsin muste_riou chronois aio_niois sesige_menou”, with one prepositional phrase nested inside the other one. (A mystery is always a revealed secret in NT Greek, btw; which certainly fits the context here.) I have a bad habit of thinking that the “eonian” refers to the “mystery”, but strictly speaking it refers to the “times”; otherwise the suffix would match {muste_riou}. (um… I think. )
Also note that there isn’t a specific preposition for the prepositional phrase, so we have to make a guess about what the preposition is; unlike {muste_riou sesige_menou} which is a special form of possessive prepositional phrase that practically never needs a Greek preposition. As it happens, the nearest prior preposition is {kata}, which can mean several things including down-from, and I think the suffixes for “times eonian” would synch up with it. (A little fuzzy there.)
So you have a fun translational option set: you can go with something like “from the revelation of a mystery hushed in never-ending times” (except those times did end); or you can go with something like “from the revelation of a mystery hushed in times which endured for an age” (except the times, plural, would seem to be more than one age); or you can go with something like “from the revelation of a mystery hushed in times which kept going for as long as those times” (now there’s some redundancy for you!); or you can go with something like “from the revelation of a mystery hushed from Godly times” or “from God’s own times” or “from times coming from God” or “in God’s own times”. Which are all admittedly kind of clunky, but they get across the idea that those times were set up by God. (And also perhaps the idea that this all has something to do with that which proceeds from God’s own essential reality. Which sounds one way or another like something to do with Christ; whom, or Whom, the revelation just happens to be about!)
To make matters worse, the whole doxology (vv25-27) has a weird habit in textual families of shifting around within Romans or even being omitted outright. So I make a point not to hang too much on this particular doxology anyway, for better or for worse.