Excellent question, LoudMusic!
On fore-and-aft rigged sailing boats, "traveller" is mostly taken for granted these days as a mainsheet traveller.
Two generations back, travellers were more common as names for things that moved inboard and outboard
or up and down. The tack of a jib
topsail on gaff-rigged cutters was hauled inboard and outboard
on a traveller, which was usually an iron (or bronze) ring encircling the bowsprit
, with a tackle to haul the traveller in- and outboard. Travellers might also be found on stays and shrouds.
In general the lines doing the hauling of the traveller were "hauls". Inhauls, outhauls, uphauls, and downhauls, depending on the context.
I agree with StuM - I've not heard anyone use a name other than "control line" or just "control" for a mainsheet traveller.
Quite thought provoking to realise (1) that no dedicated name has arisen; and (2) that we take for granted - until your query - that we lack a dedicated name.