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Having it cost stam would be wildly counterproductive since that is needed to block.
Have it cost mag and heal you for your stam and your missing health would be solid. 20% of your stam (like 7k, or 3.5k~ in PvP) and then 30% of your missing health (for PvP builds like a max of 4.5k increase) would be passable without being OP. My MDKs one unbuffed is like 7k+33% for reference. Currently GDB on my stamDK would be around 5k buffed at real low health, pretty ***.
But to be honest, it could be too OP since stam has vigor, which is really strong. Warden has both vigor+burst and can run FM, which makes it too tanky. I'd focus on making wings(+rally) a better skill, a class skill that actually has the strength to get kind of near a FM+shimmering warden or cloaking NB in defense.
The thing is that I run GDB, Wings, Volatile, and Fossilize. I have like 13.5 k Magicka and 850 Magicka recovery. It still pushes the limit. I enjoy the utility of all of these skills but it be nice if at least one of them had a Stam morph.
Stam DKs are in the regrettable situation of having only 2 of their class skills stamina based. I often run GDB on my stam DK for the buffs. But overall, stam DK has many issues in Cyrodil now.
I solved these issues by taking Thogard's advice on running a stam DK build on a warden that I leveled up. Now I mostly play the warden in PvP and I play it almost exactly the same way I played my DK, but I have more burst, a shield (wings equivalent) that works and also generates ultimate, and more healing.
Maybe Thogard can re-link his video...which was sadly completely correct...
Or, maybe they will someday rebalance and Stam DK will be stronger.