I think the answer varies not on the time, but on the circumstances involved.
For example, the Clan Robertson has two names.
Donnachaidh (children of Duncan), which is either derived from either King Duncan I (father of King Malcolm Canmore, first of the Canmore kings) or Duncan the Stout (the first chief), I forget which.
Robertson is derived from Chief Robert the Grizzled, who captured the assassins of King James I. King James II rewarded the chief with a barony, after which the clan was referred to as Robertson.

For specific clans, you might want to check their web pages - it's usually in the clan history section.