The Hebrew word for brother-in-law is גיס - gis, and sister-in-law is גיסה - gisa. Klein takes the conservative view, and writes:
Aramaic-Syriac גיסא, aphetic for אגיסא, which is of uncertain originHe was probably following Ben-Yehuda, who also wrote that the origin of the word is unclear, but then added that
there are those who say that it comes from gis גיס in Aramaic, which means "side"Gis as side is familiar from the Aramaic phrases chad gisa חד גיסא - "the one side" and idach gisa אידך גיסא - "the other side". Avshalom Kor (Yofi Shel Ivrit, page 84) agrees with this explanation, and writes that gis is a "relative on the side".
None of the modern dictionaries connect gis to gayis גייס - "troop", which is the source of giyus גיוס - "draft". For example, Klein only points out that gayis is related to the Arabic word jaysh - "army".
However, Jastrow has a more extensive etymology for gis that connects it to gayis as well. He says that both are from the root גוס, which means "to come in contact, touch, be connected, meet". The Aramaic verb גוס does mean "meet" in Pesachim 110b and Gittin 65b - I couldn't find anyone who disagreed with that. He then goes on to say that gisa גיסא meaning "neighborhood, side" derives from that root (I guess sides are connected to one another), and then connects brother-in-law to side, as we've seen before.
He connects gayis as well, by saying that it means "troop", but "especially ravaging troop, invaders, robbers". The connection here seems to be that an invader comes in contact with the area he invades.
But there is one thing I don't understand in Jastrow's theory. He also writes that the root גוס is the source of another meaning of גיס - "intimate, familiar". He brings examples from Ketubot 85b and Kiddushin 81a, where the phrase גיס ביה - "familiar with him" is used. It seems clear to me that this is just the Aramaic form of the Hebrew -gas ba גס בה. However, Jastrow says that the Hebrew gas comes from an unrelated root, meaning "to be bold", so gas ba means "he may be become bold towards her" (Ketubot 12a).
This etymology of gas isn't too far from Klein, who says that it originally meant "coarse, bulky" and then meant "vulgar, impolite, discourteous". But what caused Jastrow to miss the obvious connection between the Aramaic and the Hebrew? Was it his need to tie in gisa and gayis, and explain the meaning of gis as "meet"? Does anyone have a more charitable explanation?