This article from Mac World (for macOS 10.12.5, not Mac OS X 10.5.8) suggests that the key could have been set up to speak the selected text.
This would consistent with the "Funk" error noise each time I pressed the c key, because there was no text selected for it to read out! I tested this by selecting some text and, lo and behold, it read the text out.
The interface in macOS 10.12.5 is different enough that it's not clear where to find the settings from the article.
Open System Preferences and click "Speech".
Click "Text to Speech" in the tab bar and then, next to the "Speak selected text when the key is pressed" click the "Set Key..." button.
Chose a new key combination (mine was set to c1). I changed it to Option + Esc.
I can now type the c key!
1 It was actually set to Caps Lock + c. Why this was being triggered without Caps Lock being turned on or depressed I am unsure.