If for any reason, a judge of the 72 great sanhedrin left the great sanhedrin, who would be the one to appoint a new one?
Avraham Büchler in his book "Hasanhedrin" ("The Sanhedrin" or "Das Grosse Synedrion in Jerusalem und das Beth-Din in der Quaderkammer des Jerusalemischen Tempels" in the original German) brings some sources about the appointment of new dayanim:
"ומשם שולחין ובודקין כל מי שהוא חכם ועניו ושפל וירא חטא ופרקו טוב ורוח הבריות נוחה עליו עושין אותו דיין בעירו משנעשה דיין בעירו מעלין ומושיבין אותו בהר הבית משם מעלין ומושיבין אותו בחיל משם מעלין ומושיבין בלשכת הגזית"
Translation: "...and from there they send and check whoever is wise and humble and fears sin and the minds of people are at ease with him, shall be made a judge in his city, and once he has been made a judge in his city he shall be sent for and placed in the Temple Mount, and from there they send for him and place him in the courtyard and from there they send for him and place him in the Chamber of Hewn Stone." (Tosefta Sanhedrin 7:1, Tosefta Chagigah 2:4 and similarly in Tosefta Shekalim 3:17, Bamidbar Rabbah 6:3 and Sanhedrin 88b)
From here it seems that a person must first be made a judge in his own town. Presumably this appointment was by local leadership and/or the local court. From there, he might advance in rank and eventually reach the Great Sanhedrin in the Chamber of Hewn Stone. Examples for such cases can be seen from Rabban Yochanan ben Zakai who was a posek in the Galilee for many years before moving to Yerushalayim, according to Buchler, presumably because he was made a member of one of the higher courts and perhaps also Rabbi Yochanan ben Hachoranit1.
Shalom Albeck in his book "Batei Hadin Bimei Hatalmud" (The Courts In The Time of the Talmud) notes that the local courts' power came from the people, meaning, there were no laws to compel people to come be judged in front of the local courts. Rather, the local courts were more like halachic advisors that were also available for judicial proceedings, should cases be brought before them. As such, the appointment of these sages as judges was no different than the appointment of other religious figures within every community, as defined in the baraita in Sanhedrin 17b:
"And it is taught in a baraita: A Torah scholar is not permitted to reside in any city that does not have these ten things: A court that has the authority to flog and punish transgressors; and a charity fund for which monies are collected by two people and distributed by three, as required by halakha. This leads to a requirement for another three people in the city. And a synagogue; and a bathhouse; and a public bathroom; a doctor; and a bloodletter; and a scribe [velavlar] to write sacred scrolls and necessary documents; and a ritual slaughterer; and a teacher of young children."
The only difference between the judges and the other appointees is that appointing judges is a mitzvah. These appointments were decided by the community itself, as was the case of the appointment of Levi bar Sisi (appears in several places, such as Beresheet Rabbah 81:2) and the unnamed Babylonian scholar (Yerushalmi Shvi'it 17a).
It was often the case, however, that the local judges were the local rashei yeshiva, who had smicha. The smicha was mainly an appointment to becoming a head of a yeshiva or beit midrash and to teach students, but also gave them the ability to judge various types of court cases (Chanoch Albeck, Smicha and Minnui and Beit Din, pg. 86). As such, the appointment of these judges was naturally different from that of those described previously, as they were appointed by local sages, rather than the general community. As such, Judicial processes that came to their doorstep were more of "it comes with the territory" rather than being chosen specifically for that role by the community. An example of this can be seen in another baraita, in Sanhedrin 32b:
"The Sages taught: The verse states: “Justice, justice, shall you follow.” This teaches that one should follow the Sages to the academy where they are found. For example, follow after Rabbi Eliezer to Lod, after Rabban Yoḥanan ben Zakkai to Beror Ḥayil, after Rabbi Yehoshua to Peki’in, after Rabban Gamliel to Yavne, after Rabbi Akiva to Bnei Brak, after Rabbi Matya to Rome [Romi], after Rabbi Ḥananya ben Teradyon to Sikhnei, after Rabbi Yosei to Tzippori, after Rabbi Yehuda ben Beteira to Netzivin, after Rabbi Yehoshua to the exile [gola], i.e., Babylonia, after Rabbi Yehuda HaNasi to Beit She’arim, and after the Sages in the time of the Temple to the Chamber of Hewn Stone."
To reach the Great Sanhedrin following this path, explains Shalom Albeck, one began as a sage with smicha from one of the communities and eventually makes his way to the Sanhedrin. According to him, this process of rising in rank is hinted in the Sifrei Devarim 3462:
""And He was a King in Yeshurun"...When the Nasi places elders in the yeshiva of below, His great name is exalted above, viz. "And He was a King in Yeshurun" — When? "when the heads of the people gathered." And there is no "gathering" but that of the elders, as it is written (Bamidbar 11:16) "Gather unto Me seventy men from the elders of Israel.""
tl;dr: It seems there were two ways a member of the Sanhedrin was appointed: a. A scholar was first appointed by the community to serve as a local judge or halachic advisor/posek - from there, he might receive summons to join a higher court (presumably by the members of that court) - eventually he'll reach the level of Sanhedrin, at which point, if the Nasi saw fit, he would be summoned to join the Sanhedrin. b. A scholar would receive smicha, i.e, an appointment by his rabbi and other sages to serve as the head of a local beit midrash - from there he might move on to a greater beit midrash or to a higher court - and from there, eventually, he might receive summons from the Nasi to join the Sanhedrin. In both cases, the final say came from the Nasi, but it seems that appointees could only be deemed suitable for the position if they managed to advance in the ranks beforehand, which wasn't up to the Nasi.
1 I think he means that the name "ben Hachoranit" or "ben Hachorani" is a reference to his hometown (Choron, Beit Choron or perhaps even Choran in Syria), and not the name of one of his parents and that he was called so because he eventually came to Yerushalayim to be a member of one of the higher courts (for example, Rabbi Elazar ben Tzaddok mentions something he learned from Rabbi Yochanan ben Hachorani in Tosefta Sukkah 2:4, and we know Rabbi Elazar ben Tzaddok had many teachings about the customs of the people of Yerushalayim). And if such titles are meant to demonstrate the fact that these sages were sent as representatives of their hometowns to the higher courts, then perhaps also Rabbi Elazar Ha'Moda'i can be added.
2 This is based on the Sifrei edition published by Louis Finkelstein, which is not the version that appears in Sefaria, hence my own edits based on the Hebrew quote brought by Albeck. In the edition, on this passage, Finkelstein notes that this version of the midrash comes from Midrash Hagadol, in Yalkut Amos 548.