I wanted to describe that 「えっと」 was similar to "Ummm" in English when someone is trying to think about what they are going to say, but I wasn't even sure if that was correct. Is that just a generalization? What does 「えっと」actually come from? Can it be considered to be a word or phrase?
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えっと is related to 言えないと, which is no longer said. Over time,it's just become something we say to fill the silence when thinking, like "anno". In the Edo Times you would hear people say 言えないと, which would have sounded like, "yunaito" back then. Think of how 行かないと means "I have to go."
えっと is just a sound. Yes, it is like "umm" in English and it is used when you're thinking of what to say, hesitating, or otherwise trying to fill the silence with something before you speak. As Dono mentioned, you can find えっと's general form in the dictionary, so it can be considered a word.
There is some discussion of this on the internet here, but this discussion also boils down to it just being a sound.
えっと as just a sound is a satisfactory answer to me in terms of etymology. I'm not sure if the と in it is related at all to the quotation particle と, but I have to wonder if anyone is sure.
Basically your description of えっと as "umm" is fine, and the dictionaries define it as something that you say to think before saying something else.