Are there any Talmudic sources that back up the claim that Cyrus was really the son of Queen Esther?
The Gemara in Rosh Hashana 3b(which talks about when to start the counting of a kings reign) writes that Koresh(he was a kosher king),Daryavesh(his actual name),and Artachasasta(name of his kingdom) are one in the same.
The question is was Daryavish Esther's son. The simple answer is that its a machlokes. Tosfos on that Daf explains that Daryavish was Eshter's son based of the Midrash Va'yikra Rabah 13:5. However, Rashi holds on 4a that Daryavish was a ben noach, seemingly Rashi did not go with the opinion of the midrash.
The Kalla Rabbasi 2:15 seems to indicate this.
מה בנה של אסתר כתיב (ישעיה מה) אני לפניך אלך והדורים אישר דלתות נחושה אשבר ובריחי ברזל אגדע...
Don't know if you mean Cyrus 1 or 2 (just found out from another commenter that there was more than one), but at least one seems to have preceded the megillah: http://www.chabad.org/holidays/in-depth/default_cdo/aid/39643/jewish/Megillah-with-Commentary.htm And it came to pass in the days of Achashverosh (Esther 1:1) Achashverosh succeeded Cyrus as ruler of the Persian Empire, toward the end of the Jews' seventy-year exile in Babylon (Rashi)