There are many commentaries and midrashim that specify other people or discuss the righteous person as a broader principle (e.g., Radak and others associate this with Israel, who survived because they didn't worship idolatry).
But the idea of identifying King Jeconiah to the verse is connected to the first part of the verse. Midrash Tanhuma 4:2 specifically links the first part of the verse ("his spirit within him is puffed up, not upright") with Nebuchadnezzar (who, the midrash teaches, despite being puffed up as a king, was embarrassed by his own image because he was not upright and physically misshaped). By contrast, King Jeconiah was released after years in prison once Nebuchadnezzar died, by the successor to the throne, Nebuchadnezzar's son Avel-Marduk (cf. II Melachim 25:27). Considering this conflicting history of kings, Rashi links the Nebuchadnezzar midrash of the first part of the verse with Jeconiah as the "righteous" man of the passage, and borrows language from Vayikra Rabba (18:2) to support his comments on Avel-Marduk succeeding him.
If you're looking for clearer reasoning, see the commentary of R. Yosef Kara (who lived contemporaneously in Troyes with Rashi). For Kara, Jeconiah represents the merit of Israel in returning from Babylonian exile. Kara also understands this and neighboring verses from Chavakuk as referring to the Babylonians, and so Jeconiah is an example of the righteous man who survived--indeed, released from captivity--because of his faith during this time, just as the prophecy declared with all of the Judean exiles :
וצדיק באמונתו יחיה - הוא גלות יכניה שחזרו מגלות בבל. וכן הוא אומר בירמיה "כה אמר יי' אלהי ישראל כתאינים הטובות האלה כן אכיר את גלות יהודה אשר שלחתי מן המקום הזה ארץ כשדים לטובה. ושמתי עיני עליהם לטובה והשיבותים על הארץ הזאת ובניתים ולא אהרוס ונטעתים ולא אתוש" (כד , ה - ו). וגם שם , שהיה בגלות , נאמר עליו "וידבר אתו טובות ויתן את כסאו מעל לכסא המלכים אשר אתו בבבל" (מ"ב כה , כח).
The righteous shall live by his faith: This refers to the exiles [in the time of] of King Jeconiah, who returned from Babylonian exile. As it is said in Jeremiah (24:5-6): "Thus said the LORD, the God of Israel: As with these good figs, so will I single out for good the Judean exiles whom I have driven out from this place to the land of the Chaldeans. I will look upon them favorably, and I will bring them back to this land; I will build them and not overthrow them; I will plant them and not uproot them." And similarly (II Mel. 25:28) [when Avel-Marduk released Jeconiah from imprisonment]: "He spoke kindly to him, and gave him a throne above those of other kings who were with him in Babylon."