There has been some speculation that the woman to whom the map on the inside covers of TaT is inscribed is the daughter of Eugenides and Irene. That would be fun - keeping the stories all in the family, if you will - but I don't think it's likely.
My main reason is the princess's name. It's a good point that Eugenides's daughter might be referred to as "Kingsdaughter," analogous to how Kamet is "Kingnamer." However, Gitta does not seem like an Attolian or Eddisian name. The majority of the people from the Little Peninsula have names of Greek extraction or that simulate the phonetic construction of Greek and Latin, and the name Gitta does not fit that pattern. [examples](Irene, Sophos, Eurydice, Galen, Agape, Thalia, Janus, Helen, Heiro, Chloe, Aglaia, Xanthe, Philologos, Hilarion, Ochto, Gorgias, Iolanthe, Xenophon, Eugenides, Erondites, Ambiades, Xorcheus, Aristogiton, Stenides, Temenus, Sejanus, Teleus, Relius, Lamion, Laecdomon…) Some names, like Costis, Pol, Tarra, and Brinna, deviate from the Greek motif, but Gitta isn't similar enough to them for me to be persuaded that the princess is Attolian.
The name "Gitta" does have a Spanish or Italian sound to it (Margita, Margherita, Gitana), but it could also be Scandinavian/Germanic (Brigitta) or Indian (Gita). I propose three possible countries this mystery princess could be from: the Braels, Ferria, or a yet-unnamed analogue to India.
At first glance, I thought "Gitta" looked Mede, mainly because TaT was set in the Mede empire, but also because of the letters and sounds involved. Although the name is not sibilant like Nahuseresh, Senesh, Sudesh, Gessiret, or Akretenesh, the 't' sound could correspond with the 't' in Melheret, Kamet, and Simkit. However, scrutiny proved a Mede princess highly unlikely: the Medes are ruled by an emperor, not a king, and they do not think highly of women as rulers.
The case for the princess being a Braeling lies mainly with her second name: Scandinavian countries historically, and Iceland still today, use patronyms rather than surnames. Girls take their father's name or title with "-dottir" at the end; boys take their father's name with "-sson."
TaT is the first book to introduce the Braels, and they are patently Scandinavian. The Braeling who Kamet met on the way to Perf was named Skell or Skerrell, both Norse-esque names; he had pale skin and fair hair, mentioned having cut off a thick beard, and referred to his familiarity with cold weather. The ambassador's name, Yorn Fordad, also sounds Norse, and the one named allied ship, the Hammer of Yeltsever, is like a reference to Thor. The Braels played a large part in dealing with the secret navy - and that indicates that they have a good-size navy of their own. Vikings were known for their prowess on the sea, and the Braelings may have a similar proclivity.
Also, in Norse culture, women were highly regarded and in many cases were equal with men (as early as the eighth century, Viking women could, for instance, own property, inherit property and authority, and divorce their own husbands - things that weren't allowed in most of the world until the nineteenth or twentieth century). The fact that the princess is being educated points to some cultural recognition of women's worth, which aligns with the notion that she could be from the Braels.
MWT never includes anything lightly, so after the subtle but key position the Braels held in the political drama in TaT, it stands to reason that the country will play a part in the next book; they're not just going to disappear offstage. If Eugenides is establishing a close relationship between Attolia and the Braels, then it very well may be that the Braeling princess is being educated about her country's political allies and their history.
The case for the princess being Ferrian lies mainly in the Italian sound of "Gitta." Just as Attolia, Eddis, and Sounis are very like Greece and Turkey, Ferria seems much like Italy. We don't have much data regarding their names besides Malatesta, Sophos's Ferrian tutor, but that name alone makes a very good case for Ferria being analogous to Italy: the House of Malatesta was an Italian noble family in the thirteenth to fifteenth centuries. And, according to Wikipedia, their motto is "The Indian elephant isn't afraid of mosquitos" (Elephas indus culices non timet). That is a strange motto for an Italian family to have, but given what MWT has said about the next book having something to do with elephants, it seems worth bearing in mind. As others have pointed out, Dite is in Ferria and may well be spying for Eugenides. These factors could be setting up for Ferria to enter the stage significantly in the next book.
That hint about elephants leads to the third possibility for the princess's nationality. There isn't much to support an Indian analogue except for the facts that Gita is an actual Indian name and that one species of elephant is the Indian elephant. The name Gita means "song" or "sacred poem / sacred song," which fits conceptually with MWT's propensity for including within the main story another layer of stories that parallel aspects of the plot. However, the Indian subcontinent is far from Turkey (Sounis), Persia (the Mede empire), Greece (Attolia, Eddis), Italy (Ferria), France (the Gants, the Pents), and Scandinavia (the Braels), so unless the next book is another wide-ranging travel tale, I think an Indian-analogue princess is unlikely.
Given the cues in the books so far, which country do you think the princess is most likely from? Or do you have another reason to think that the map really is for future progeny of our favorite queen and annux?