What is the etymology and definition of הגיס?
(Reference Rashi in Chulin 108A ד״ה טיפת חלב שנפלה על החתיכה and ד״ה ניער את הקדרה)
Rashi (Beitzah 34a) says that it means to stir (literally overturn) with a spoon:
מגיס - מנער בכף, שקורין קומיבר"א
Rashi (Shabbos 18a) explains similarly that it means to overturn.
מגיס - מהפך בה, ובמבושל הוי בישול.
This is clear from context in a number of places.
Edit: As @ba pointed out, this word is Hebrew, not Aramaic as I assumed. You can read my original analysis below the line, and my newer update here.
Jastrow has this word under G-O-S, which means "come into contact with," and as a result of that, הגיס means "stir". I'm not sure how one follows the other, but okay.
In finding which Hebrew word this relates to, I can only suspect that it is the same as N-G-S (concordance results here), which seems to mean "to pressure" (in line with the above). I would move more to a meaning of agitation in this case, although I don't know that it will necessarily fit across the board. Feedback welcome.
However, it seems like quite the mystery as to how/why it means that, and I would like to suggest that it is a combination of two words that are both spelled similarly.
"Gas" appears to mean two or three different things (in many of the lexicons, including Jastrow). One meaning is some sort of serving dish - and not a spoon. This can be found in places such as Targum Onkelos as a translation of Keara (Bamidbar 7, for all of the Nesi'im):
קׇרְבָּנ֞וֹ קַֽעֲרַת־כֶּ֣סֶף אַחַ֗ת
קוּרְבָּנֵיהּ מְגִסְּתָא דִּכְסַף חֲדָא
The root seems to be from M-G-S, and CAL gives us the Peshitta/Syriac to Bamidbar 4:7 (also translation of Keara) as the source:
mgs, mgsˀ (magsā) n.m. vessel 1 a type of plate or jar Syr. P Nm4:7 : ܘܢܪܡܘܢ ܒܗ ܡܓ̈ܣܐ ܘܟܦ̈ܐ ܘܡܢ̈ܩܝܬܐ ܘܛܪ̈ܘܠܐ ܕܢܘܩܝܐ they should put down dishes and spoons, and libation bowls and spoons . IshNum 88:8 : ܡܓ̈ܣܐ ܠܦܝ̈ܠܣܐ ܩܿܪܐ܂ ܒܪܡ ܦܝ̈ܠܣܐ ܫܛܝܚܝܢ ܡܢ ܡܓ̈ܣܐ .
I can only imagine that the source of this word is N-G-Sh in Hebrew, which means to serve, and therefore, some sort of serving platter is related to this word (technically it means to come/bring forth, but is used to mean serve, such as Bereishis 27:25).
Another usage seems to be to "come up", rise, as per Targum Pseudo-Yonasan to Devarim 17:20:
לְבִלְתִּ֤י רוּם־לְבָבוֹ֙ מֵֽאֶחָ֔יו
מטול דלא יגיס לביה יתיר מן אחוי
CAL has a number of similar words, with G-S-Y meaning a fountain, to vomit up, and other similar words, all of which relate to bringing up. This likely also comes from the same root as above as G-Sh is used a number of times to mean come forward/come up.
I would suggest that the action of mixing got it's name from a combination of these. The mixing action was likely to lift the food inside the pot, and therefore, one becomes a Megis. However, it is possible that since the spoon was also serving portions of the food, this name stuck as well, even though it originally referred to other vessels.