Today I finally made progress on accessing the Thunder Forge. I wish I had a firm grasp of the true progression, but the test realm has been forcibly progressed by the developers and I'm not 100% certain of the player requirements. We were in Stage 5 of the unlocking process until today and I don't know if this quest is available sooner than that. From what I've read in comments on Wowhead, this quest chain was not implemented until this past week or so, which is when Stage 5 was already active. I also have no way of knowing whether there are rep requirements for accessing this quest, but I am currently Honored with the Kirin Tor Offensive.
It may well be that Stage 5 is required, that is when I completed the new scenario for the Thunder Forge. I don't know whether you'd be able to reach the Thunder Forge area until after you finish that quest (done via scenario). For now I would hedge your bets and make sure you do every non-daily quest that Jaina offers you.
It's possible that you could just find Itoka (Master of the Forge, blue dot) outside the Thunder Forge and kill him. He drops the Strange Metal Ingot you need to start the quest. You can find him at the marked location on this map:
After you get his item, you enter the Thunder Forge room (red dot) and there will be a book laying by the forge that you can click to complete the quest. Your reward will be a Lightning Steel Ingot and Notes on Lightning Steel. The note contains seven new recipes: one for making a Lightning Steel Ingot and six plans for "Reborn" weapons. The six plans are Drakefist Hammer, Reborn | Fireguard, Reborn | Lionheart Blade, Reborn | Lunar Crescent, Reborn | Planar Edge, Reborn | Thunder, Reborn. Each of these recipes is still orange at 600 skill, even though they are marked as learnable at 500 skill.
I went ahead and learned the Lightning Steel recipe and made one - as suspected it does indeed have a daily cooldown. This will put a pretty effective throttle on the creation of Lightning Steel and limit how many Reborn weapons show up for sale. That's at least 4 days for a rare weapon and 14 days for epic weapons. If the tooltips are to be believed, you'll need to make each rare weapon in order to learn the epic version. Then you'll need to make the epic version to learn the upgraded epic version of that. This seems to model the original path of progression for these recipes where you were able to upgrade each weapon twice to get a more powerful version.
You also learn new recipes when making lightning steel. Today I learned the recipe for the Haunted Steel Greathelm. Apparently the item for this recipe is not yet implemented. I will post additional recipes as I learn them from the daily cooldown. I also plan to run my Engineer through this quest and report back on that process. Feel free to leave questions in the Comments!