Function generator output level drifting
In case this is useful to others with similar problems in future, I opened up the function generator (power off and unplugged) and gently poked a few things, especially the output level potentiometer. Since then, the output level was dead stable for 4 hours yesterday and again for two hours this morning, so it seems to have fixed it, for now at least.

Link to other suggestions people made in addition to those above, with schematic and photo with the cover off.

As a follow up two weeks on... 

BK Precision recommended first checking the output level potentiometer (spot on, it has been stable ever since) and have since dug around in their files and provided a component list with zoomed in sections of the schematic. It is a pretty ancient machine (schematic is from 1994), so all in all I say brilliant customer service from BK Precision!