What do I want for the Mongols in CK2?
1) A succession system that utilizes character traits and game rules to create potential succession crises. Since EU: Rome and Vae Victus already do something similar, I have confidence we will see something like this. In CK1, the only nod to these issues were events, and they were never satisfactory.
2) Early diplomatic options: First contact with the Mongols might not be the appearance of 150,000 soldiers in your lands, but instead be merchants and or nomads crossing borders. It should be possible to forestall invasion or even prevent it if you can keep cordial relations with the Khans (avoiding Khwarazm's mistake, anyone?)
3) Submission should be possible: You should be able to become a tributary of the Mongol Empire, even if you are a king. This should cost considerable money and prestige, but the option is sorely needed. Playing for time should allow for players to pull a Muscovy when the time comes.
4) Religious conversion for Mongol rulers should be possible in-game: No more pagan Mongols for the whole game just because you started in 1066. Religious conversion should also help legitimize Mongol rule and let them play religious political games. It should also result in civil wars and assassinations, like it did historically.
5) No more super-armies: There should be an end to 150,000 man armies that suffer no attrition and that fight equally well in mountains and steppe. This was, perhaps, a necessary evil in CK1, but I always hated that mechanic.
6) The Mongols need some kind of game mechanic that makes large-scale on-map unification difficult to achieve and maintain. In CK1, it is too easy for one horde to annex the other horde and create a super horde. Historically, stuff like that required Ghengis, Kublai, or Timur like abilities. Large hordes should only stay large and unified when a great leader is in power. When only mediocre leaders are in power, you should see smaller hordes fighting each other as well as their neighbors.
7) There should be more diplomatic options between Mongols and non-Mongols. Alliances should be possible, even when religious differences are present. The historical Mongols allied with Christian states like Bulgaria and even gave Venice and Genoa trade rights in places like Kaffa. The Mongols were not space aliens that killed everyone in sight and leveled all structures; they played politics and diplomacy, too.