In the discussions of Mashiach, there is a very prevalent idea of there being a coming "savior". Are there any references to this idea in Tanakh, and if so, what are they?
There are two clear references to a single person being a "Moshiach" which brings Israel to salvation and through that, the people recognize Gd as the ultimate source of redemption.
However, because of the nature of the prophets, a single verse can not be quoted and be understood properly.
See here in Isiah, the end of chapter 44 and the beginning of chapter 45, where Karush (King Cyrus) is called Gd's Moshiach.
See here in Yechezkel were the descendant of King david, is marked as a single king over a united Israel:
However, the idea that the Moshiach does magic, and must be a perfect person, is not defined in Tanach. The Moshiach of Tanach is a regular political leader, who brings about the rebuilding of the Temple, the teshuva of the Jewish people, and the unification of Israel.