According to Klein, the root קמץ means "to enclose with the hand, grasp, take a handful, close, shut". By a switch of the labial consonants, we find two other related verbs with similar meanings: קפץ and קבץ, to which Klein adds כוץ as well. He also connects it with the Aramaic קמעא, meaning a little, a handful. It also might be connected to קמיע kamia - an amulet, related to the Arabic "qama'a (= he tamed, curbed, bridled.)" Steinberg goes further and connects קמץ and the other verbs above to a series of words beginning with the same two letters and having related meanings: קמט - "to grasp" or "to wrinkle", and kemach קמח - flour ground fine and small.
From קמץ we get a number of interesting words. A kamtzan קמצן is a miser, who holds his hand tight. Jastrow provides us with three small animals named kamtza קמצא - a locust (the related קפץ means "to hop"), an ant ("scraper, collector") and a snail (which Jastrow feels might be an error for לימצא, and Steinsaltz on Shabbat 77b says that comes from the Old French limace, meaning slug or snail.)
There are some drashot that connect the name Kamtza, from the famous story of Kamtza and Bar Kamtza, with kamtza meaning locusts. However, Steinsaltz (Gittin 55b) says the name Kamtza derives from the Greek kompsos, meaning elegant and refined, but also with a more negative meaning - crafty.
According to some, Medieval Latin camisia is a borrowing through Late Classical Greek kamision from the Central Semitic root קמץ represented by Ugaritic qms ('garment') and Arabic qamis ('shirt'). From camisia we get such English words as camisole and chemise.
[Others, however claim the development worked in the opposite direction: that Arabic qamis is derived from the Latin camisia (shirt), which in its turn comes from the Proto-Indo-European kem ('cloak').]
Another derivative of קמץ is the vowel kamatz (or qamatz). According to Horowitz (pg. 56):
The verb קמץ means to draw together. The Ashkenazim pronounce the קמץ with lips drawn together. That's how the vowel got its name. If the men who wrote our present niqud pronounced the קמץ as do the Sephardim, "ah", they would certainly have never called it קמץ.
While he doesn't mention it, I assume that explains the vowel פתח patach (open) as well.
As a final note, to return to the subject of shirts, Stahl writes that there were those who suggested calling a T-shirt in Hebrew a hultzat kamatz חולצת קמץ (based on the T shape of the kamatz), but it never caught on...