I hadn't put a great deal of thought into this topic thus far, but now that I know that we're likely to only see two start dates at launch, I'm going to have to refine my opinions somewhat, because 1066 simply isn't enough for me. It has its high points, certainly, but I wouldn't consider it the most interesting or definitive start date by any means. People naturally gravitate toward what they consider to be the default or most natural choice: if the Third Crusade was treated by the game as the primary start date and placed first in the list, I figure odds are decent that we'd all be talking about how much people enjoy playing Richard the Lionheart and Saladin, but that's neither here nor there. If need be, I'll be the very first in line to mod the 12th-14th centuries into the game, because I honestly believe they're a more representative and enjoyable sandbox to play in, but of course I'd much rather see them implemented with the same degree of care and professionalism that the rest of the game will enjoy, if not at launch, then as an expansion down the road.
So, start dates. My primary criteria is really quite simple: what dates have the largest number of interesting characters and situations available to explore? There are a few that come to mind. The dates aren't particularly specific, and are weighted more to enjoy the largest number of overlapping reigns belonging to noteworthy characters that we would be inclined to start as.
1102, slightly after the First Crusade, because this gives a foothold in the Holy Land to Raymond IV, count of Toulouse, in establishing himself in the future county of Tripoli, last of the mainstream early crusader states. Baldwin I reigns in Jerusalem, the future Baldwin II in Edessa, and Bohemond I in Antioch. Henry I has just recently taken the throne in England, and Poland is divided in civil strife after the death of Wladyslaw I. Your archetypal crusader start.
Then we get a little bit more interesting. 1131: Fulk and Melisende ruling in Jerusalem. The magnificent Roger II just crowned in Sicily, and Henry I drawing close to the end of his life in England, with the stage set for the fallout over Matilda's inheritance, and a young Eleanor of Aquitaine. Alfonso the Emperor ruling over Leon, Castile, and Galicia, as well. The early crusader states coming close to the peak of their historical power and influence.
1189, the Third Crusade almost speaks for itself. Richard the Lionheart. Saladin. Philip Augustus. Frederick Barbarossa. Rǫgnvaldr Guðrøðarson, king of the Isles, "the greatest warrior in the western lands." William the Good, king of Sicily, and William the Lion, king of Scotland. Conrad of Montferrat. Balian of Ibelin. Guy of Lusignan. The iconic Crusader Kings era, when almost all the famous names and dynasties are in play.
1205-1206. Two words: Genghis Khan. But also the aftermath of the Fourth Crusade. King John in England. Valdemar the Conqueror, king of Denmark. Also important to capture in this window of opportunity: Boniface of Montferrat, king of Thessalonica. The first start that would offer a playable kingdom of Cyprus. Alexios and David Komnenos founding Trebizond. The emperor Kaloyan in Bulgaria. One of the first two Latin Emperors in Constantinople.
1248, for similar reasons to 1189. Right before the Seventh Crusade. Saint Louis and that famous adventurer, Charles of Anjou. Henry III, king of England, Simon de Montfort, and Llywelyn the Last in Wales. Frederick II Hohenstaufen. James the Conqueror, King of Aragon. William of Villehardouin, prince of Achaea. Birger Jarl in Sweden. Baibars as a young commander.
There are other dates I could suggest, with equally fascinating casts of characters, especially later ones, but for now I'll settle for the battles that I might at least have a modest chance at winning.