How did Aristotle or St. Thomas Aquinas (such as in one of his commentaries on Aristotle) resolve the liar's paradox?
Aquinas doesn't have anything to say about the liar's paradox, to my knowledge (informed, but finite) knowledge. However, a slightly later medieval philosopher named John Buridan certainly does.
Take a look at Gyula Klima's paper on the topic to get a sense of his solution here. In essence what Buridan does is say that the Liar sentence is false, but then he attempts to block the inference that falsity of the Liar sentence entails that the negation of the Liar sentence is true. See Klima's paper for details on how this is supposed to work.
This is the closest I've been able to find in St. Thomas, where he discusses whether truth is immutable (Summa Theologica I q. 16 a. 8 ad 3):
A proposition not only has truth, as other things are said to have it, insofar, that is, as they correspond to that which is the design of the divine intellect concerning them; but is said to have truth in a special way, insofar as it indicates the truth of the intellect, which consists in the conformity of the intellect with a thing. When this disappears, the truth of an opinion changes, and consequently the truth of the proposition. So therefore this proposition, "Socrates sits," is true, as long as he is sitting, both with the truth of the thing, insofar as the expression is significative, and with the truth of signification, insofar as it signifies a true opinion. When Socrates rises, the first truth remains, but the second is changed.
And from his Quæstiones Disputate De Veritate a. 6, on whether truth is immutable:
The truth of a thing is the cause of the truth of a proposition, for a statement is said to be true or false insofar as a thing exists or does not exist.
it is essential to a proposition that it signify that which it has been made to signify.
Thus, according to St. Thomas, the liar's paradox "propositions" are not real propositions at all because there is no corresponding thing for them that "exists or does not exist" and they do not "signify that which" they have "been made to signify."
Aristotle's Sophistical Refutations said "The argument is similar too concerning the same man's lying and speaking the truth at the same time" lead an Archbishop of Canterbury, Thomas Bradwardine (c. 1300–1349), to write his Insolubles at Oxford between 1321 and 1324.
“Insolubilia and Bradwardine's Theory of Signification.” Medioevo: Revista di storia della filosofia medievale, 7: 115–34. claims:
For the opinions mentioned above were those of the old [logicians], who understood little or nothing about insolubles. After them there arose the prince of modern philosophers of nature, namely Master Thomas Bradwardine. He was the first one who discovered something worthwhile about insolubles.