Wikipedia has a good explanation of the origin of the Clyde name:
'This surname originated in pre-Celtic times, and was used in Scotland and Ulster as a name for people who lived by the banks of the River Clyde, which flows through Glasgow.
The origin of the river's name comes from the gaelic name "Cluaidh", the original meaning of which is now unclear. Some believe it is derived from the Brythonic (pre-Celtic) word "clut", meaning "the cleansing one". There was also a Celtic goddess named Clota, who was considered the guardian of the river, and the Roman name for the river was Clota.'