I live in the south west of England and there are many villages and roads that feature the word "Clyst". For example, Broadclyst, Clyst St Mary, Clyst Honiton and so on.
What does clyst mean, and where does it come from?
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Clyst is the name of the river and all the localities you cite are on its banks.
Presumably all the Clyst St xxx are named after the saints to whom the various churches were dedicated. The villages being named after the churches.
Broadclyst is when the valley or the river itself broadens downstream.
Clyst is a Celtic name meaning 'clear stream'.(the Devon shire is named after the Celtic Devon tribe that was living there before the Roman conquest), just like Avon or Thames.
Other Celtic words that made it into the English toponymy are "crag", meaning "rock", "coombe", meaning "deep valley" (also in French), "tor" meaning "peak", "car-" (remember Harold's Carlisles) meaning "a fortified place".
I would say the meaning is clear stream as clyst can be traced back to 937ad and is derived from hhtor meaning clean.