The basis of the idea that Prester John was from Central Asia had to do with the fact that many Central Asian nomads, Mongols included, were Christian (usually Nestorian) during the period of the Mongol conquests into Europe and the Middle East.

So, you can forgive European Christians the idea that there was this Christian leader somewhere to the East, who would put the hurt on all the Muslims they'd been crusading against. There was no internet, news and rumor spread by word of mouth, and the fact that there really were Central Asians, who were Christians, fighting Muslims...well, you can see how that fits right in to the whole Prester John story.

Genghis Khan, though, wasn't a Christian. You could say he was a Shamanist, or worshiper of Tengri, the Mongolian sky god. But among his grandchildren and especially their wives and mothers, there were quite a number of Christians. These were usually Kerait or Naiman (and the Naiman, incidentally, like most other Mongolized Turkic tribes, eventually converted to Islam. Although not very successfully, as I point out to my Naiman wife, who sneaks bacon sandwiches on a regular basis).

If you're asking if "Prester John" is related to the name "Genghis" and the title "Kahn", then no. They share no roots, since Kahn is a Mongolic (or at least central Asian) title for a warlord and "Genghis" is a corruption of "Chinggis". "Prester" is a corruption of the Greek word presbyteros (elder, priest). John is... well, John. The names and titles are not etymologically linked.

Now, if you're wondering if the person Ghengis Kahn and the legendary Prester John were conflated, that's a different story. In the past (and in the earliest stages of the crusades) Western monarchs attempted to contact Prester John to ask for reinforcements against the Muslim empire, calling on their shared Christianity. We now know that these missions failed. However, when Ghengis Kahn began conquering eastern Muslim holdings, some Christian leaders initially interpreted the news as Prester John springing to life. Of course in time the Christian West would come to learn the truth about the Mongols, particularly when missionaries were sent to them. According to the records of some Dominican friars, at least one particular encampment of Mongols believed Christians worshiped "rocks and trees". This was clearly not the Prester John they had hoped for.

So, the wikipedia article seems to have a pretty good discussion on this topic.

The link between Prester John and Genghis Khan was elaborated upon at this time as the Prester became identified with Genghis' foster father, Toghrul, king of the Keraits,... Joinville describes Genghis Khan in his chronicle as a "wise man" who unites all the Tartar tribes and leads them to victory against their strongest enemy, Prester John. William of Rubruck says a certain "Vut", lord of the Keraits and brother to the Nestorian King John, was defeated by the Mongols under Genghis. Genghis made off with Vut's daughter and married her to his son, and their union produced Möngke, the Khan at the time William wrote. According to Marco Polo's Travels, the war between the Prester and Genghis started when Genghis, new ruler of the rebellious Tartars, asked for the hand of Prester John's daughter in marriage. Angered that his lowly vassal would make such a request, Prester John denied him in no uncertain terms. In the war that followed, Genghis triumphed and Prester John perished.

The historical figure behind these accounts, Toghrul, was in fact a Nestorian Christian monarch defeated by Genghis. He had fostered the future Khan after the death of his father Yesugei and was one of his early allies, but the two had a falling out. After Toghrul rejected a proposal to wed his son and daughter to Genghis' children, the rift between them grew until war broke out in 1203. Genghis captured Sorghaghtani Beki, daughter of Toghrul' brother Jaqa Gambu, and married her to his son Tolui; they had several children, including Möngke, Kublai, Hulagu, and Ariq Böke.

You said:

I was under the impression that the Prester John myths predated Crusader knowledge of the Mongols,

This is also true:

Though its immediate genesis is unclear, the legend of Prester John drew strongly from earlier accounts of the Orient and of Westerners' travels there. Particularly influential were the stories of Saint Thomas the Apostle's proselytizing in India, recorded especially in the 3rd-century work known as the Acts of Thomas. This text inculcated in Westerners an image of "India" as a place of exotic wonders and offered the earliest description of Saint Thomas establishing a Christian sect there (the Saint Thomas Christians), motifs that loomed large over later accounts of Prester John

The legend moved around quite a bit, it also landed Prester John in Ethiopia, which was (and largely remains) Christian, which was the aspect of the story I had been most familiar with.