What is the root and source of the term "sanhedrin," the name of the Jewish high court and of a tractate in the Talmud?
As far as I know, it is not from scripture.
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Etymologically, Sanhedrin is a late Hebrew representation of the Greek word synedrion συνέδριον meaning "sitting together" as a legislative assembly or Senate.
Jastrow supports Yishai's answer, that "סנהדרין" derives from the Greek συνέδριον:
סַנְהֶדְרִין, סַנְהֶדְרֵי
f. (also pl.) (συνέδριον) Sanhedrin, the supreme council of the Jews; ס' גדולה the Great S., consisting of seventy-one members; ס' קטנה the Small S., a judicial court of twenty-three. Snh. I, 6. Ib. ראויה לס' fit to be a seat of the S. Macc. I, 10 ס' נוהגת בארץ וכ' the S. may exercise its functions in Palestine and outside. Ib. ס' ההורגת וכ' a S. that executes capital punishment (more than) once in seven years. Ib. 9 (ref. to Deut. XVII, 6) שלא תהא ס' שומעת וכ' this intimates that the S. must not hear the testimony from the mouth of an interpreter. Snh. 63a לס' שהרגו וכ' that a S. which puts a person to death must not taste food during the entire day of execution; a. v. fr. - Pl. סַנְהֶדְרָאוֹת, סַנְהֶדְרָיוֹת, סַנֶּדְרָ'. Ib. I, 5 אין עושין ס' לשבטים וכ' supreme courts for tribes (provincial courts, Small Sanhedrin) can be instituted only by decree of the court of seventy-one; Sifra K'dosh. ch. VIII, Par. 10 סנדריות של ישראל; Yalk. Lev. 619 סנהד' של שבטים, opp. ס' גדולה. Snh. 16b סנדראות (also in Chald. diction); a. fr. - Sanhedrin, name of a treatise, of the Order of N'ziḳin, of Mishnah, Tosefta and Talmud Babli a. Y'rushalmi.סַנְדְּרִי, סַנְהֶדְרִין, סַנְהֶדְרִי
ch. same. Targ. Y. II Num. XXV, 4. Targ. Y. I ib. 7. Targ. I Chr. XVIII, 17; a. fr. - Snh. 16a; a. fr. - Pl. סַנְדַּרְיָיתָא, סַנְהֶדְרְיָיתָא. Targ. Y. I Lev. XXIV, 12. Targ. Y. I Num. IX, 8; a. fr.
As user6591 mentioned in his answer, the word is used in Mishnayos Sanhedrin, 1:6 as "סנהדרי." Tiferes Yisrael, there (#43) writes that this is the Latin (לאטיין) word for it. He also says that the rabbis of the Mishna chose to use this word, because it (is נוטרריקון) stands for "שונא הדרת דין."
The actual words of the Maharil are found in the Likutei Maharil #6 אמר דלכך נקראו סנהדרין, מפני ששונאין הדרת פנים בדין, נוטריקון שלו כך. This roughly translates as they do not show favoritism while judging.
I too was always bothered as to the origins of this word, until I saw the following Midrash Lekach Tov on Parshas Beha'aloscha: ומהו לשון סנהדרין, סין זה תורה שניתנה מהר סיני, הדרין שמהדרין התורה במדרשה ומיפים ומישרין הכתוב זה עם זה-"What is the meaning of the term 'Sanhedrin'? 'Sin' refers to the Torah which was given from Mount 'Sin'ai, 'hadrin': since they beautify (מהדרין) the Torah in its extrapolation, and make better and straighten out that which is written, one (part) with another."
In maseches Sanhedrin chapter one, mishna 6, the word is presented as sanhedrei. The Tiferes Yisroel #43 quotes the Aruch who writes this is the Latin version of the name. He also quotes the Maharil as saying Chazzal chose this word because in notrikin it stands for Sonei Hadras Din.
The actual words of the Maharil are found in the Likutei Maharil #6 אמר דלכך נקראו סנהדרין, מפני ששונאין הדרת פנים בדין, נוטריקון שלו כך. This roughly translates as they do not show favoritism while judging.
Other answers have already noted that this comes from the Greek συνέδριον ("synedrion"). I'll add more detail: That word comes from the Greek σύνεδρος ("sitting together"), from σύν ("with", also found in e.g. English synergy and synchronize) and ἕδρα ("seat", also found in English cathedral and distantly related to English sit).