Actual Tannaim in the Sanhedrin

Question

I know Shimon "HaTzadik" was in the Keneset Hagdola, but it's not the same.

Do we know by name any Tannaim who actually sat at the Great Sanhedrin of 71 sages?

Answer 1

The 2nd Mishna in the 2nd Perek of Chagiga gives a list of the last 5 sets of Av Bet Din and Nasi of the Sanhedrin before the second Churban - for a total of 11 Tanaim; members of the Sanhedrin whose names we know.

יוֹסֵי בֶּן יוֹעֶזֶר אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, יוֹסֵי בֶּן יוֹחָנָן אוֹמֵר לִסְמוֹךְ. יְהוֹשֻׁעַ בֶּן פְּרַחְיָה אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, נִתַּאי הָאַרְבֵּלִי אוֹמֵר לִסְמוֹךְ. יְהוּדָה בֶּן טַבַּאי אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, שִׁמְעוֹן בֶּן שָׁטָח אוֹמֵר לִסְמוֹךְ. שְׁמַעְיָה אוֹמֵר לִסְמוֹךְ. אַבְטַלְיוֹן אוֹמֵר שֶׁלֹּא לִסְמוֹךְ. הִלֵּל וּמְנַחֵם לֹא נֶחְלְקוּ. יָצָא מְנַחֵם, נִכְנַס שַׁמַּאי. שַׁמַּאי אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, הִלֵּל אוֹמֵר לִסְמוֹךְ. הָרִאשׁוֹנִים הָיוּ נְשִׂיאִים, וּשְׁנִיִּים לָהֶם אַב בֵּית דִּין:‏

Yose ben Yoezer says that [on a festival] the laying of the hands [on the head of a sacrifice] should not be performed.
Yose ben Johanan says that it should be performed.
Joshua ben Perachia says that it should not be performed.
Nittai the Arbelite says that it should be performed.
Judah ben Tabai says that it should not be performed.
Shimon ben Shetach says that it should be performed.
Shamayah says that it should be performed.
Avtalyon says that it should not be performed.
Hillel and Menahem did not dispute. Menahem left, Shammai entered.
Shammai says that it should not be performed.
Hillel says that it should be performed.
The first members of each pair served as Nasi, and their counterparts served as Av Beis Din (deputy Nasi) of the courts..

Answer 2

Following @Danny's answer, More sages that held the titles, therefore been in the great sanhedrin:

Nasi:
It could be that Shimon Ben Shetach was a Nasi, and not Av Beis Din.
Shimon Ben Hilel
Raban Gamliel hazaken (the elder)
Raban Shimon Ben Gamliel hazaken (the elder)
Raban Yochanan Ben Zakai
Raban Gamliel of Yavne*, part time in rotation with Rabbi Elazar Ben Azaria
*Also Av Beis Din

Av Beis Din:
Rabbi Yehoshua Ben Hanania

After this era, there are more sages carrying the title, but it could be that the authority of the sanhedrin was impaired, so It wasn't a fully operating sanhedrin:

Nasi:
Raban Shimon Ben Gamliel II
Rabbi Yehuda H-Nasi
Raban Gamliel III
Rabbi Yehuda Nessia
Raban Gamliel IV
Rabbi Yehuda Nessia II
Hilel Nessia
Raban Gamliel V
Rabbi Yehuda Nessia III
Raban Gamliel VI

Av Beis Din:
Rabbi Natan
Rabbi Efes
Rabbi Hanina bar Hama

So, 7 more names for sure, maybe even 20.