What does "Clyst" mean?

Solution 1:

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".

Solution 2:

Wikipedia and the Oxford Dictionary of British Place Names agree that it's a Celtic word, but disagree about its meaning ("clear stream" vs "sea inlet").