The tenth letter of the Hebrew alphabet is yod (or yud or yodh). The origin of the letter's name is clear - it looks like a yad יד - arm or hand (see here and here.) It is the smallest of the Hebrew letters, and due to its size the Greek version of the name - iota - came to mean "a very small amount" and from iota came the word jot, which later meant "to make a short note of."
Yad is a very common word in the Tanach - Even Shoshan gives 1617 listings. So it is a bit beyond the scope of this post to deal with every nuance of the word. Besides hand or arm, it can also mean: handle, stem (of a fruit), monument / place, power / strength, part / portion, side, pointer (some Biblical examples here.) It is part of a large number of prepositions in Hebrew:
- ביד - by, through
- כיד - according to the power of
- ליד - near
- מיד - immediately
- לידי - to
- מידי - from
- על יד - next to
- על ידי - through
But while Haketav Vehakabbalah tries to explain how vidui is different than hodaah הודאה - praise (which would seem to be from the same root), Steinberg connects all three meanings. First of all, he writes that vidui is identified with bowing, prostrating oneself, throwing oneself on the ground - as in Nechemia 9:3 - מִתְוַדִּים וּמִשְׁתַּחֲוִים and Ezra 10:1 - וּכְהִתְוַדֹּתוֹ, בֹּכֶה וּמִתְנַפֵּל.
He then goes on to write that the verb הודה - to give praise, thanks (and the root of todah תודה - "thank you") - was also connected with bowing. As a proof, he shows how the Targum translates Shmuel II 16:4 - הִשְׁתַּחֲוֵיתִי as מודינא. He writes that only once in the Tanach is הודה used towards a person - יְהוּדָה, אַתָּה יוֹדוּךָ אַחֶיךָ - "Yehuda, your brothers will praise you" (which is a play on words on the name Yehuda.) In all other places, the verb is used towards God, where bowing and prostration are very appropriate. This concept was carried forward in to the Amida prayer, where we bow when we say מודים אנחנו לך.
From the Daat Mikra on Yoel 4:3, it would seem that perhaps there is a connection between ידה - to throw away - and נדה - the root of nidah נידה and nidui נידוי, both referring to a person cast away. But it's not entirely clear from that source, and the other sources I've checked are ambiguous at best. (For example, Klein says that נדה is related to Akkadian nadu ("to throw"), but doesn't connect it back to ידה.)
Lastly, just in case you were wondering, Philologos shows that yad is not connected to "yadda, yadda, yadda."