Just as a warning, this is pure speculation and I have no Idea what it will actually be, these are just my thoughts.
Something that struck out to me most imminently about the X0-1 Power armor was it's blatant visual relations to Advanced power armor of the Fallout 2 era Enclave. (probably to partially please fans who absolutely hated the dog-face armor from fallout 3) So, it's obvious it's related to the Enclave in some way, the question is how. After finishing up Broken Steel, Elder Lyons remarked that this might not be that last we see of the Enclave, but it's pretty obvious we've broken their power structure. Which could lead to them seeking new places and ways to control the wasteland. Somewhere that is on-par with them technologically and equally want to bring order to the wasteland. Where else close by is like that other than The Institute :)
That is correct, I am (possibly wildly) speculating that some remnants of the Enclave joined up with The Institute, but how does that relate to Power Armor you ask? The Institute has a lot of resources for research and development, and if their new former-Enclave buddies wanted some new armor, they would be one of the few places still able to produce it. You might just think it could be just an outdated Enclave Power armor but consider this. Why is it called X0-1? "-1" would imply it's the first of it's kind, when in Fallout shelter it is noted as "The most advanced piece or armor created by human hands." Which according to Fallout 3 Hellfire is the latest and most advanced to be produced by the Enclave. meaning that this armor is newer than Hellfire. And it couldn't be just another new armor because of the visual similarities to the Advanced power armor, which it is clearly a derivative of.
So because it has a new naming convention different from previous Enclave themed armors but still bears the trademark look and is actually newer than Hellfire, I believe that this was produced for Enclave remnants of some kind by the Institute.
I've probably gone too far but I'd like to know what you guys think.