The Dragon's Potential Heads
With there being three dragons it's logical to assume there will eventually be three dragon riders. There is one already -- Daenerys, who rides atop her largest and most fearsome dragon, Drogon (named for her late husband). From birth, Drogon has been closest to Daenerys and by far the most protective. That bond, though tested when Drogon would fly free across Meereen and beyond, has only become stronger, and Daenerys' control over Drogon has increased. For evidence, just look back on the last two episodes of Game of Thrones.
Debate still rages over whether Daenerys is the prince(ss) that was promised, and if it will be she who finally sits upon the Iron Throne, but there is no denying that she has reclaimed her Valyrian and Targaryen heritage - bringing dragons back to world, and specifically, to Westeros. She is quite obviously one of the dragon's three heads, if not the head, following in Aegon the Conqueror's footsteps. All she needs now are the two who will take the place of Aegon's sister-wives.
So, where does this leave Jon? Well, that Targaryen heritage implies he could well have "the blood of the dragon", a link back to the first dragon riders of Valyria. Now, it should be noted that not all Targaryens inherently possess this (just look at Viserys and his golden crown), and Jon certainly isn't fireproof since he burnt his hand back in season 1 while rescuing the Lord Commander from a wight. But this Jon is a different Jon, and it's quite possible his resurrection has changed him in more ways than he (or we) can yet know.
In this week's episode, "Eastwatch," Game of Thrones showed us the most convincing piece of evidence yet for why Jon could be the next to ride a dragon. When Daenerys is returning to Dragonstone she lands just feet away from Jon, who had been watching her and Drogon's arrival from the cliffs. Drogon approaches Jon, at first baring his teeth and snarling, but as Jon reaches out his hand the dragon calms and actually allows Jon to touch him. In that moment it's evident Drogon is reacting to something about Jon, be it his Targaryen lineage or to demonstrate that he's someone Daenerys can truly trust -- possibly even so far as to let him ride one her dragons. (Which, if that happens, it'd have to be Rhaegal, right? It's too poetic an opportunity to ignore).
Picking a third head for the dragon isn't nearly as easy, but the favorite among most fans is Tyrion Lannister. There are several explanations for why Tyrion is a potential dragon rider, each with varying strengths of evidence to support them. A popular one is that Tyrion isn't really a Lannister but secretly also a Targaryen, with the thinking being that Aerys (The Mad King) raped Tywin Lannister's wife, Joanna, and is therefore Tyrion's true father. It isn't impossible, and would certainly explain why Tywin hated Tyrion so much, but it also seems a bit lazy for the story to contain not one but two secret Targaryens. Plus, author George R.R. Martin has stated that being Targaryen isn't a prerequisite to being a dragon rider, and in fact, during the Dance of the Dragons several non-Targaryens bonded with and rode dragons.
The best piece of evidence in Tyrion's favor is that outside of Daenerys and now Jon, he is the only person to have gotten up close and personal with the dragons and not been burnt alive. This was during the period when Tyrion was governing Meereen in Daenerys' absence and out of sheer curiosity, Tyrion enters the crypt below the Great Pyramid where Viserion and Rhaegal are chained up and frees them. It's an incredibly moving scene as Tyrion has his greatest dreams come to life before his very eyes, even managing to touch one of them briefly before removing their chains. It isn't really clear which dragon it is he touches and first releases, and to be fair the other dragons haven't received anywhere near as much screen time as Drogon, but our best guess is Viserion judging by its golden hue (though again, that might just be the torchlight).
Whether this brief interaction confirms Tyrion as the third head of the dragon is up for debate, and the scene is arguably not as compelling as the moment Jon experienced in "Eastwatch." But then, dragon taming and riding is a lost art much like forging Valyrian steel, and what actually goes into it isn't really known. It's said that a dragon only ever allows one person to be their rider at a time, and that they remain bonded to that person for as long their rider lives. It's possible, then, that what happened to Tyrion below the Great Pyramid was the first step towards him forging a bond with one of Daenerys' dragons, leading to him becoming another dragon rider.
Not only does the scene seem to back this up, but Tyrion's strong interest and respect for the dragons may play into creating a bond with them. Dragons are intelligent creatures, and it's clear they sense that Tyrion is no threat and possibly even that he's a friend of their mother. They may also sense his desire to know dragons and appreciate his kindness towards them -- a sentiment made abundantly clear with that dear story of him crying himself to sleep when told that all the dragons were gone. Game of Thrones is rarely a story of wish fufillment, but it'd certainly be a cathartic moment were Tyrion to receive the dragon he so wished for as a child.
There are other potential candidates, of course, and each come with their own qualifications. One possibility is Bran, who with his ability to warg could enter one of the dragons and become dragon rider that way. Then there's Jorah, arguably the character closest to Daenerys and therefor someone her dragons should trust as implicitly as she does. And to make things only more confusing, there's a character in the novels called Young Griff who claims to be Rhaegar's son, Aegon, who Varys is said to have whisked out of King's Landing before he could be killed. If his claim turns out to be true, he would be a Targaryen and strong candidate to bond/ride with a dragon.
In the end, there is no definite answer for who are the dragon's three heads, nor is there any reason the phrase need apply to who will ride Daenerys' other two dragons. In both A Song of Ice and Fire and Game of Thrones, prophecies never have clear cut answers and rarely do they play out as we think they should. It should also be noted that while the television series has introduced the Azor Ahai/prince that promised prophecy, it's made little to no mention of the dragon having three heads, suggesting it may not be as important as fans like to believe. Regardless, the moment shared between and Jon and Drogon in this week's episode and the earlier scene between Tyrion, Viserion, and Rhaegal are sure to carry some significance, either as a sign that these are true allies Daenerys can trust or of something more.