I personally find British panel shows to be among the funniest shows you can find on TV. One common occurrence I've noticed is that the panelists are often seen writing scribbling notes while another member is talking.
In particular I've noticed it on the shows QI and Have I Got News for You, where some panelists seem the write furiously in between making jokes or answering questions. That said t's certainly not limited to just shows.
One guess is that, given the unrehearsed nature of those shows, they are writing down jokes as they occur to them, to be made a a later more appropriate moment. Still the shear consistency at which pen and paper is provided, and panelists are seen writing, suggests there must be a well established reasoning behind it beyond my random musings, but search for a semi-official explanation has never turned up any results.