Once you spawn, survivor of traitor does not matter, exit the cabin and walk directly to the map. It will zoom in and if you hold A you will receive your bonus objective.
Some examples I have received:
Craft 3 electrical parts. (1 stone and 1 wood)
Craft 3 fuel. (2 berries each)
Eat 3 pot pies. (1 berries and 1 meat)
Craft 2 first aid kits. (3 herbs cooked)
Eat 3 cooked meats. (Gather raw meat from moose or wolves)
Craft 2 fire bundles. ( 2 wood each)
Use radio to call rescue.
There have been more, if you get a particularly hard bonus objective please share in the comments.