I haven’t really found how they did it back on the Starship, but I found a little article that is quite interesting and e.g. mentiones how they solved the lightning thing back then.
http://www.aviatorservices.com/starship_history_1.htm

Seems like today they do it with an aluminum mesh. And then there is the question of conductivity of carbon fibre vs say boron or aramid fibre composites.

PS: Do you know what happens when older composites are hit by lightning? They are blown to pits and pieces because they tend to accumulate moisture within the structure and that turns to steam and blows apart the material.