THAT'S ACTUALLY A GREAT QUESTION and it boils down to a few things in particular. at least part of it is the simple fact that she reminds him of his home. the castle he grew up in is located in the north and there's pretty much snow year round there, as well as the fact that she's a queen, but those are almost superficial? i think what's more important is that she's been the most straight-forward and helpful of all the personae he's had a chance to speak with and that counts for a lot with him. because he's tried to talking to a few others, but their conversations have always been brief and nnnnnnnot particularly helpful. but they had a fairly productive conversation that inspired him to speak with beauty and learn more about the darkness. and then there's also the fact that as a knight, jon is fairly firmly planted in the "i want to help the personae as best i can" camp, but, for the most part, he doesn't really know how to do so outside of games. and while he's happy to play in them, he also doesn't feel particularly adept at a lot of them, mostly because he's still so new! so he's looking for a tangible way to help them and she told him about her lost heart, which is a very tangible thing he can look for/quest for.
the final thing, i guess, is that even though jon doesn't know a lot about patronning, he tends to think of as being the equivalent of a lord/lady-vassal type relationship. and so when he looks for some to patron with, he's very specifically looking at it through that lens, so he's looking for someone that he wouldn't mind serving as a knight might serve their lord. and while he likes beauty well enough, she doesn't quite ping him in that way. even though he's only had a couple conversations with the snow queen, there's something about her personality that clicks well with jon. she strikes him as the sort of person he can grow to understand more about and his instincts tell him that she's not a bad or deceitful person, which makes him want to speak with her more and eventually take her as his patron. i think the fact that she's somewhat cold and dispassionate specifically pings him because that plays directly into his idea of what a lord/lady/queen/king ought to be like and, thus, his idea of what he'd want out of a patron.
and i'll reverse the question! why would the snow queen consider taking jon on as her knight?
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the final thing, i guess, is that even though jon doesn't know a lot about patronning, he tends to think of as being the equivalent of a lord/lady-vassal type relationship. and so when he looks for some to patron with, he's very specifically looking at it through that lens, so he's looking for someone that he wouldn't mind serving as a knight might serve their lord. and while he likes beauty well enough, she doesn't quite ping him in that way. even though he's only had a couple conversations with the snow queen, there's something about her personality that clicks well with jon. she strikes him as the sort of person he can grow to understand more about and his instincts tell him that she's not a bad or deceitful person, which makes him want to speak with her more and eventually take her as his patron. i think the fact that she's somewhat cold and dispassionate specifically pings him because that plays directly into his idea of what a lord/lady/queen/king ought to be like and, thus, his idea of what he'd want out of a patron.
and i'll reverse the question! why would the snow queen consider taking jon on as her knight?