On Wikipedia, citing "The Celtic Place-Names of Scotland" by William J. Watson (first published 1926), "The derivation is not certain but probably from the Brythonic Clouta which became Clut in Old Welsh. The name's literal meaning is 'wash' but probably refers to the idea of a river goddess being 'the washer' or 'strongly flowing one'."Also on Wikipedia, this proposed Brythonic root is translated as "the cleansing one".
Unfortunately none of those suggestions appear to be true. The "loud" meaning is clearly folk etymology (i.e. made up) by someone trying to understand a Brythonic word by comparing it to a more comprehensible Old English word (hlod). The old British word for the Clyde however is Clut, modern Welsh Clud and Gaelic Chluaidh. Unfortunately there don't appear to be any real words even close in form in Welsh, Irish or Gaelic. The best suggestion I've seen is that it is the name of the local river goddess, and possibly pre-dates the spread of Celtic into Britain. At this point, as with many ancient names, it is impossible to say what it means, or satisfactorily verify any theory.
Found this possible definition, which makes great sense, at the following-- http://wiki.answers.com/Q/What_does_the_name_'Clyde'_mean"Clyde is Scots-Gaelige for: Heard from afar. The river Clyde means a loud river heard from a distance. The Firth (of Clyde) means a fjord-like deep inlet (not a shallow delta) to the river near the sea. Clywwd, is also welsh-Brythonic for: Loud or loudly. Clwwd is Cornish perhaps for: warm. These Gaeilige, Welsh, and Cornish words were "Anglicized" by Germanic (Angle Saxon Dane) invaders into Britain. Gaelic languages are from an Indo-European stock becoming distinct in Galatia (in modern Turkey) before; Persian administration, Greek trade and Roman colonialism, among other influences moved the languages and peoples ever westward into Gaul (modern France). These languages included what would later evolve into Brythonic (Brittany-Gaul-France-Britain) Cornish-(Cornwall and southern Britain) Welsh (Wales) Gaelige (Eire-Ireland Gaelic-Scots-Pict)."
Also on Wikipedia, this proposed Brythonic root is translated as "the cleansing one".