The word dagesh is first found in the medieval works of Hebrew grammarians (follwoing the Tiberian Masoretes). It was not borrowed from Aramaic (at least not the Aramaic of the Targumim, or Babylonian and Jerusalem Talmuds), but from the related Syriac language. In Syriac, the verb דגש means "to pierce", and has a cognate in the Akkadian dakasu (see the entry here in the Chicago Akkadian Dictionary, and the article included here - "Use of Akkadian DKS") of the same meaning.1
We do find a Babylonian Aramaic form of dagesh - דיגשא digsha, but this is later (from the Babylonian Masoretes, who placed the dot above the letter), and not found in Talmudic literature.
Why "pierce"? Apparently, this is due to the dot "piercing" the page, and we find a similar relationship between the English (and earlier Latin) words "punctuation" and "puncture".
Gesenius, in his Hebrew Grammar (page 55), suggests that since dagesh might have also meant "to sharpen", perhaps the word was chosen not because of the mark in the letter but due to the "sharpening" of the sound:
The root דגשׁ in Syriac means to pierce through, to bore through (with sharp iron); hence the name Dageš is commonly explained, solely with reference to its form, by puncture, point. But the names of all similar signs are derived rather from their grammatical significance. Accordingly דגשׁ may in the Masora have the sense: acuere (literam), i.e. to sharpen a letter, as well as to harden it, i.e. to pronounce it as hard and without aspiration. דָּגֵשׁ acuens (literam) would then be a sign of sharpening and hardening (like Mappîq מַפִּיק proferens, as signum prolationis), for which purposes a prick of the pen, or puncture, was selected.
It is unclear to me whether Latin grammar influenced this choice of a word, or whether they developed in parallel. I had the same question here about the relationship between the word geresh גרש and "apostrophe", and I still have not found an answer.
Today the verb דגש in the hifil form - הגדיש hidgish - means to emphasize or highlight anything, not just a consonant. But this usage is very new - it doesn't even appear in Ben Yehuda's dictionary.
1. I did find two sources that seem to preserve this earlier meaning.
a) Targum Yonatan on Mishlei 12:18 translates יֵשׁ בּוֹטֶה כְּמַדְקְרוֹת חָרֶב "there is one who speaks like the piercing of a sword" as אית דאמריה ספסירא רגשא. But the Syriac translation (the Peshitta) translates it as אית אמרין ספסירא דגשא. My guess is that the Targum רגשא is a misreading of the more logical דגשא - "piercing".
b) The Torah Shleimah here quotes the medieval collection of midrashim Sechel Tov saying that God "מכה ומדגיש כל גאה ורם" - "smites and madgish מדגיש the proud and haughty" - I assume that madgish here means to stab, or better, the Syriac meaning "to beat".)