Quite a few suggestions in the last seven pages that I would agree with (and so won't echo here), but most of you have unfortunate and unfathomable taste 
* A(nother) surviving Liao dynasty, or failing that, an enduring Jin dynasty. What's the appeal? Mongols and Tatars are just soooooo over. And personally the Ming dynasty lasting longer holds all the appeal of a case of malaria lasting longer. I actually have played around with such timelines (e.g., a different faction of the terminal Liao court strikes off for the northeast instead of toward the west), but more for conlang purposes than alt-history ones.
* For that matter, what about more durable para-Chinese states like the Xia ('Tangut') or Nanzhao kingdoms?
* A Far Eastern Republic that wasn't folded up after the Russian Civil War would be a fertile ground for alt-historical political weirdness, I imagine.
* The Yezo Republic! Even if you only let it survive ten or twenty years, it would be weird and fun.
* Of course, any Sumerian, Elamite, or Harappan polity should have lasted much longer. (Mostly so that we might have more philological materials to work with, I have to admit.)
* A(nother) surviving Liao dynasty, or failing that, an enduring Jin dynasty. What's the appeal? Mongols and Tatars are just soooooo over. And personally the Ming dynasty lasting longer holds all the appeal of a case of malaria lasting longer. I actually have played around with such timelines (e.g., a different faction of the terminal Liao court strikes off for the northeast instead of toward the west), but more for conlang purposes than alt-history ones.
* For that matter, what about more durable para-Chinese states like the Xia ('Tangut') or Nanzhao kingdoms?
* A Far Eastern Republic that wasn't folded up after the Russian Civil War would be a fertile ground for alt-historical political weirdness, I imagine.
* The Yezo Republic! Even if you only let it survive ten or twenty years, it would be weird and fun.
* Of course, any Sumerian, Elamite, or Harappan polity should have lasted much longer. (Mostly so that we might have more philological materials to work with, I have to admit.)