The term "מותא תנינא" (or מותא תניינא) (Mota Taneina), Aramaic for "second death", appears in a few verses translated by Targum Jonathan, traditionally ascribed to the late-1st century BCE-early 1st century CE sage Jonathan ben Uziel:
"אֲמַר נְבִיָא בְאוּדְנִי הֱוֵיתִי שְׁמַע כַּד אִתְגְזַר דָא מִן קֳדָם יְיָ צְבָאוֹת אִם יִשְׁתְּבַק חוֹבָא הָדֵין לְכוֹן עַד דִי תְמוּתוּן מוֹתָא תִנְיָנָא אֲמַר יְיָ אֱלֹהִים צְבָאוֹת."
"The prophet said, with mine ears I was hearing when this was decreed from before the Lord of hosts, namely, that this your iniquity shall not be forgiven you till you die the second death, said the Lord, the God, the God of hosts." (source)
"הָא כְתִיבָא קֳדָמַי לָא אֶתֵּן לְהוֹן אַרְכָּא בְּחַיַיָא אֱלָהֵן אֲשַׁלֵם לְהוֹן פּוּרְעֲנוּת חוֹבֵיהוֹן וְאֶמְסוֹר לְמוֹתָא תִּנְיָנָא יַת גְוִיַתְהוֹן."
"Behold, it is written before me: I will not give unto them prolongation in this life; but I will recompense unto them the wages for their sins, and deliver their bodies to the second death." (source)
"וְתִשְׁבְּקוּן שׁוּמְכוֹן לְקַיָמָא לִבְחִירִי וִימִיתְכוֹן יְיָ אֱלֹהִים מוֹתָא תִנְיָנָא וּלְעַבְדוֹהִי צַדִיקַיָא יִקְרֵי שְׁמָא אוֹחֲרָנָא."
"And ye shall leave your name for a curse to my chosen: for the Lord God shall slay you with the second death, and call His righteous servants by another name." (source)
"אַיְתָא עֲלֵיהוֹן עָקָא וִיהוֹן דָמָן לְרַוְיָא בְּדִיל דְלָא יְהוֹן תַּקִיפִין וִימוּתוּן מוֹתָא תִנְיָינָא וְלָא יֵחוּן לְעַלְמָא דְאָתֵי אֲמַר יְיָ."
"When they become heated up, I will serve them their banquet and make them drunk, that they may become jubilant and may die a second death and live in the World to Come," declares the LORD." (source, with emendations by me)
"וַאֲרַוֵי רַבְרְבָהָא וְחַכִּימָהָא שׁוּלְטָנָהָא וְטוּרְנָהָא וְגִבָּרָהָא וִימוּתוּן מוֹתָא תִנְיָנָא וְלָא יֵיתוּן לְעַלְמָא דְאָתֵי אֲמַר מַלְכָּא יְיָ צְבָאוֹת שְׁמֵיהּ."
"I will make her princes and her wise men drunk, her governors, her prefects and her mighty men, that they may die a second death and not go to the World to Come," Declares the King, Whose name is the LORD of Hosts." (source, with emendations by me)
Amnon Shapira in his essay in Hebrew "מותא תנינא - המות השני", Sinai 109, pp. 5-15, discussed the meaning of the term. First, he notes possible parallels in the terms "כלת עולמים" from a few of the Dead Sea Scrolls, "אש עולמים" from The Testament of Zebulun and "אבדון עולם" from The Psalms of Solomon, two apocryphal works. Analysis of all of the sources (as well as examples from 2nd century CE and later Targums such as Onkelos) shows that the term may have had a few meanings: a. Losing a place in the resurrection. b. Being cursed to be eternally lost. c. Being eternally annihilated. As can be seen, all three meanings are closely related.
He then suggests a conceptual parallel in the words of Beit Shammai in Tosefta Sanhedrin 13:1:
"בית שמאי אומרים שלשה כתות הן אחת לחיי העולם הבא ואחת לחרפות לדראון עולם...אבל המסורות והאפיקורסין והכופרין בתורה ופורשים מדרכי צבור ושאין מודים בתחיית המתים וכל מי שחטא והחטיא את הרבים כגון ירבעם ואחאב ושנתנו חיתתם בארץ חיים ושפשטו ידיהם בזבול גיהנם ננעלת בפניהם ונדונין בה לדורי דורות...שאול כלה והם אינם כלים..."
"...Beit Shammai say: Three groups live on in the World to Come and one is cursed forever...but the [informers?]1 and the heretics and those that deny the Torah and dissent from the ways of the public and do not believe in the resurrection and whoever sinned and caused others to sin such as Jeroboam and Ahab and caused fear in the land of the living and attacked the Temple, Gehinnom is locked before them and they are judged to be there forever and ever...Sheol becomes extinct and they do not extinct..." (my translation)
In short, the term "second death" circa the 1st century CE referred to some form of post-death punishment. What this punishment was exactly was open to interpretation. Shapira also noted that some sources seem to view it as as an active punishment while others as a passive punishment. About how well-known the term was, Shapira notes (p. 15, n. 29) that some traditional Jewish sources from the Middle Ages and onwards, as well as some modern scholars postulate that the Aramaic Targums reflect the beliefs of the common Jews of circa the time of authorship of each of the Targums, so that means that the term מותא תנינא (second death) was very well-known.
1 I'm not entirely sure how to translate מסורות. I thought it might be a feminine form of מוסרים, informers. I know it should be מוסרות, but it might be scribal error.