The term appears in some other texts. According to Meir Benayahu, "The Ashkenazi community of Jerusalem 1687—1747" [Hebrew], Sfunot 2 (1958), p. 173, n. 48, it's not an acronym but rather a made-up word based on the verse in Tehillim 42:9:
"יוֹמָם יְצַוֶּה ה' חַסְדּוֹ וּבַלַּיְלָה שירה [שִׁירוֹ] עִמִּי תְּפִלָּה לְאֵל חַיָּי"
"By day may the LORD vouchsafe His faithful care, so that at night a song to Him may be with me, a prayer to the God of my life."
It seems to come from the word "צרה", misfortune. Originally, the term seems to have been part of the phrase "ובלילה ציר"ו", which explains the connection to the verse (see e.g. the text that Benayahu discussed there; for more examples, search Otzar Hachochmah for the word ציר"ו).
As @N.T. suggested in a comment, the Chida's usage of the term within the phrase "ויהי ציר"ו" may have been based on the verse in Melachim 1:5:12:
"וַיְדַבֵּר שְׁלֹשֶׁת אֲלָפִים מָשָׁל וַיְהִי שִׁירוֹ חֲמִשָּׁה וָאָלֶף"
He composed three thousand proverbs, and his songs numbered one thousand and five.
The same phrase was also used later by Rabbi Yosef Chaim [A]buchbut1 in a eulogy he wrote for a woman named Rachel Asudari (can be found in the book Me'oraot Yosef, p. 178).
It should be noted that many made-up words or Jewish transliterations of foreign words are written in Medieval and Modern-era sources with apostrophes or quotation marks, as well as on occasion also "real" words.2 Sephardic rabbis in particular also used this form to denote words whose original meaning changed in a particular context (e.g. בְּעַרֶ"ב היא באה, as opposed to בָּעֶרֶב הִיא בָאָה).3
1 Both spellings exist.
2 For a short note on this phenomenon, see here [Hebrew].
3 I heard this example from Prof. Yaron Harel.