...and the results are in! Ranked in order of most votes, Costis's 3 books are:
Some kind of treatise about swordfighting. This was this clear winner, which actually surprised me, given that Costis doesn't seem to be super into like, technique at the beginning of KOA when he is sparring with Gen and acting annoyed about having to go through basic numbered forms! That said, I can see this as the kind of thing he could have been given as a fancy gift, maybe when he first joined the queen's army as a trainee, or maybe to recognize his accomplishments when he got picked for the queen's guard a year later
Financial advice for dummies. I included this one as a joke and am glad so many of you saw my vision! Shoutout to @inziladun for her comments/tags about whether Costis received this book before or after the Lovely Girl incident (my personal vote is that it was a pointed gift after the fact, or, possibly, that Costis bought it for himself in a fit of self-improvement after his paychecks get reinstated BUT I am still laughing at the possibility she raised that Costis's family bought it for him when he moved to the city i.e. long before anyone scammed him).
History of Attolia (Klimun/Kathodicia optional). I included this one because I thought it might be the kind of book Costis would have studied with his tutor. Plus, if we're in a world that is newly using the printing press, something like an authoritative (state-sanctioned?) history of the country seems like a plausible text for Attolia's royal press to print lots of (reasonably) affordable copies of!
Runner up: Collected Works of Melinno and/or Archilochus. This got almost as many votes as options 2 and 3, and in some ways it's the book we have strongest canonical evidence of, since Costis was required to memorize the poems he mentions studying. Maybe it was all oral tradition memorization, but the way he talks about it made me picture kid!Costis studying lines on a page. For this one, I'm not picturing a printed book, but a handwritten scroll, maybe a fairly detailed/long one, created by Costis's tutor for purposes of his education. I think it probably has "greatest hits" of Attolian literature, maybe some of them excerpted from larger works. (This is the option I voted for.)
Honorable mention to @whocalledhimannux's brilliant suggestion that one of the books could be a book of jokes, inspired by Costis's desperately awkward attempt to tell Kamet some jokes in the lion's den scene in TAT. I'm dying about this and I wish it had occurred to me to put it in the poll outright! Something about the way Costis awkwardly just starts listing jokes by heart really matches the image I now have of him buying a cheaply printed, flimsy pamphlet of jokes from some street seller in Attolia town....they're all pretty bad but they probably seemed funnier when he and Aris were taking turns reading them out loud to each other over a bottle of wine...