Because Hezekiah was a descendant of David, I doubt it is possible to completely separate this particular Messiah figure from the Davidic Messiah himself. But it seems worth pointing out that there seems to have been a mild, limited expectation that the Messiah would be Hezekiah in some way:
Babylonian Talmud, Sanhedrin 94a: 94a .... Of the increase of his government and peace there shall be no end. R. Tanhum said: "Bar Kappara expounded in Sepphoris, 'Why is every mem in the middle of a word open, whilst this is closed?' The Holy One, blessed be He, wished to appoint Hezekiah as the Messiah, and Sennacherib as Gog and Magog; whereupon the Attribute of Justice said before the Holy One, blessed be He: 'Sovereign of the Universe! If Thou didst not make David the Messiah, who uttered so many hymns and psalms before Thee, wilt Thou appoint Hezekiah as such, who did not hymn Thee in spite of all these miracles which Thou wroughtest for him?' Therefore it [= the mem] was closed. Straightway the earth exclaimed: 'Sovereign of the Universe! Let me utter song before Thee instead of this righteous man [Hezekiah], and make him the Messiah.' So it broke into song before Him, as it is written, 'From the uttermost part of the earth have we heard songs, even glory to the righteous.'" ....
Babylonian Talmud, Sanhedrin 98b: 98b .... R. Giddal said in Rab's name: The Jews are destined to eat [their fill] in the days of the Messiah. R. Joseph demurred: Is this not obvious? Who else then should eat — Hilek and Bilek? This was said in opposition to R. Hillel, who maintained that there will be no Messiah for Israel, since they have already enjoyed him during the reign of Hezekiah. .... What is his [the Messiah's] name? The School of R. Shila said: His name is Shiloh, for it is written, until Shiloh come. The School of R. Yannai said: His name is Yinnon, for it is written, His name shall endure for ever: e'er the sun was, his name is Yinnon. The School of R. Haninah maintained: His name is Haninah, as it is written, Where I will not give you Haninah. Others say: His name is Menahem the son of Hezekiah, for it is written, Because Menahem [‘the comforter’ ], that would relieve my soul, is far. The Rabbis said: His name is ‘the leper scholar,’ as it is written, Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him a leper, smitten of God, and afflicted.
Babylonian Talmud, Sanhedrin 99a: 99a .... R. Hillel said: There shall be no Messiah for Israel, because they have already enjoyed him in the days of Hezekiah. R. Joseph said: May God forgive him [for saying so]. Now, when did Hezekiah flourish? During the first Temple. Yet Zechariah, prophesying in the days of the second, proclaimed, Rejoice greatly, O daughter of Zion, shout, O daughter of Jerusalem, behold, thy king cometh unto thee! he is just, and having salvation, lowly, and riding upon an ass, and upon a colt the foal of an ass.
Babylonian Talmud, Berachoth 28b: 28b .... When Rabban Johanan ben Zakkai fell ill, his disciples went in to visit him. When he saw them he began to weep. His disciples said to him: "Lamp of Israel, pillar of the right hand, mighty hammer! Wherefore weepest thou?" He replied: "If I were being taken today before a human king who is here today and tomorrow in the grave, whose anger if he is angry with me does not last for ever, who if he imprisons me does not imprison me for ever and who if he puts me to death does not put me to everlasting death, and whom I can persuade with words and bribe with money, even so I would weep. Now that I am being taken before the supreme King of Kings, the Holy One, blessed be He, who lives and endures for ever and ever, whose anger, if He is angry with me, is an everlasting anger, who if He imprisons me imprisons me for ever, who if He puts me to death puts me to death for ever, and whom I cannot persuade with words or bribe with money — nay more, when there are two ways before me, one leading to Paradise and the other to Gehinnom, and I do not know by which I shall be taken, shall I not weep?" They said to him: "Master, bless us." He said to them: "May it be [God's] will that the fear of heaven shall be upon you like the fear of flesh and blood." His disciples said to him: "Is that all?" He said to them: "If only [you can attain this]! You can see [how important this is], for when a man wants to commit a transgression, he says, 'I hope no man will see me.'" At the moment of his departure he said to them: "Remove the vessels so that they shall not become unclean, and prepare a throne for Hezekiah the king of Judah who is coming."
Palestinian Talmud, Berachoth 17b: 17b .... And this [following story] supports the view of R. Yudan son of R. Aybo. Once a Jew was plowing and his ox snorted once before him. An Arab who was passing and heard the sound said to him, “Jew, Jew. Loosen your ox, and loosen your plow [and stop plowing]. For today your Temple was destroyed.” The ox snorted again. He [the Arab] said to him, “Jew, Jew. Bind your ox, and bind your plow. For today the Messiah-king was born.” He said to him, “What is his name?” [The Arab replied,] “Menahem.” He said to him, “And what is his father’s name?” He [the Arab] said to him, “Hezekiah.” He said to him, “Where is he from?” He said to him, “From the royal capital of Bethlehem in Judea.”
Rabbi Hillel seems to have been a realist or even a pessimist, proposing that Hezekiah was indeed the Messiah: but only in a single, already accomplished incarnation.
Rabban Johanan ben Zakkai, on the other hand, seems to have expected a future Hezekiah figure.
This expectation leads me to wonder about a certain individual mentioned by Josephus:
Josephus, Wars 1.10.5 §204-205: 204 Now Herod was an active man, and soon found proper materials for his active spirit to work upon. As therefore he found that Hezekiah, the head of the robbers, ran over the neighboring parts of Syria with a great band of men, he caught him and slew him, and many more of the robbers with him; 205 which exploit was chiefly grateful to the Syrians, insomuch that hymns were sung in Herod's commendation, both in the villages and in the cities, as having procured their quietness, and having preserved what they possessed to them; on which occasion he became acquainted with Sextus Caesar, a kinsman of the great Caesar, and president of Syria. / 204 Ὁ δὲ ὢν φύσει δραστήριος ὕλην εὐθέως εὑρίσκει τῷ φρονήματι. καταλαβὼν οὖν Ἐζεκίαν τὸν ἀρχιλῃστὴν τὰ προσεχῆ τῇ Συρίᾳ κατατρέχοντα μετὰ μεγίστου στίφους αὐτόν τε συλλαβὼν ἀποκτείνει καὶ πολλοὺς τῶν λῃστῶν. 205 ὃ δὴ μάλιστα τοῖς Σύροις ἡγεῖτο κεχαρισμένον: ὑμνεῖτο γοῦν ἀνά τε τὰς κώμας καὶ ἐν ταῖς πόλεσιν Ἡρώδης ὡς εἰρήνην αὐτοῖς καὶ τὰς κτήσεις ἀνασεσωκώς. γίνεται δ᾽ ἐκ τούτου καὶ Σέξτῳ Καίσαρι γνώριμος ὄντι συγγενεῖ τοῦ μεγάλου Καίσαρος καὶ διοικοῦντι τὴν Συρίαν.
Josephus, Antiquities §14.9.2 159-160: 159 But that youth of his was no impediment to him; but as he was a youth of great mind, he presently met with an opportunity of signalizing his courage; for finding that there was one Hezekiah, a captain of a band of robbers, who overran the neighboring parts of Syria with a great troop of them, he seized him and slew him, as well as a great number of the other robbers that were with him; 160 for which action he was greatly beloved by the Syrians; for when they were very desirous to have their country freed from this nest of robbers, he purged it of them. So they sung songs in his commendation in their villages and cities, as having procured them peace, and the secure enjoyment of their possessions; and on this account it was that he became known to Sextus Caesar, who was a relation of the great Caesar, and was now president of Syria. / 159 Βλάπτει δὲ οὐδὲν αὐτὸν ἡ νεότης, ἀλλ᾽ ὢν τὸ φρόνημα γενναῖος ὁ νεανίας ἀφορμὴν εὑρίσκει παραχρῆμα εἰς ἐπίδειξιν τῆς ἀρετῆς. καταλαβὼν γὰρ Ἐζεκίαν τὸν ἀρχιλῃστὴν τὰ προσεχῆ τῆς Συρίας κατατρέχοντα σὺν μεγάλῳ στίφει, τοῦτον συλλαβὼν κτείνει καὶ πολλοὺς τῶν σὺν αὐτῷ λῃστῶν. 160 σφόδρα δὲ αὐτοῦ τὸ ἔργον τοῦτο ἠγάπησαν οἱ Σύροι: ποθοῦσι γὰρ αὐτοῖς ἀπηλλάχθαι τοῦ λῃστηρίου τὴν χώραν ἐκαθάρευσεν. ὕμνουν γοῦν αὐτὸν ἐπὶ τούτῳ κατά τε κώμας καὶ κατὰ πόλεις ὡς εἰρήνην αὐτοῖς παρεσχηκότα καὶ ἀσφαλῆ τῶν κτημάτων ἀπόλαυσιν. ἐγένετο δὲ διὰ τοῦτο καὶ Σέξστῳ Καίσαρι γνώριμος ὄντι συγγενεῖ τοῦ μεγάλου Καίσαρος καὶ διέποντι τὴν Συρίαν.
Was this bandit chief really given the name Hezekiah at birth, or did he take it on as a nickname pointing to his role as the expected Hezekiah? Was he perhaps really named Hezekiah, from which name he drew eschatological inspiration and set out to fulfill his destiny? Or is it just a silly coincidence?
A rather late passage:
Sefer Zerubbabel: That despicable man said to me: “Zerubbabel!? What business do you have here? Who has brought you here?” I responded and said: “A wind from the Lord lifted me up and carried me to this place.” He said to me: “Do not be afraid, for you have been brought here in order that He might show you (and then you in turn might inform the people of Israel about everything which you see).” When I heard his words, I was consoled and regained my self-composure. I asked him, “Sir, what is the name of this place?” He said to me, “This is mighty Rome, wherein I am imprisoned.” I said to him, “Who then are you? What is your name? What do you seek here? What are you doing in this place?” He said to me, “I am the Messiah of the Lord, the son of Hezekiah, confined in prison until the time of the End.” When I heard this, I was silent, and I hid my face from him. His anger burned within him, and when I looked at him (again), I became frightened.