The only 'gift' we leave is a refrigerator/freezer thermometer in the fridge so new tenants can get the fridge and freezer set right especially after we turn it to barely on when the unit is empty for a bit. We had a lot of 'fridge doesn't work' calls only to find the fridge and freezer set to warmest setting and tenants pretending to be ignorant of how to use a fridge. Leaving the thermometer seems to make encourage folks to adjust it themselves. Since every tenant has taken the thermometer with them, I guess they are gifts that we give out.
If a unit is empty for a bit and a new tenant is not lined up, I usually buy a dollar store toilet brush and plunger if I find a unit doesn't have one when turned over. That way if there is an accidental clog I'm prepared and there is a brush handy to do a quick clean on the toilet if needed. I leave those in place along with making sure there is a roll or two of toilet paper and paper towels to dry hands. I leave the flyers or menus for the local pizza and chinese food places that I know are good/reliable (a lot of pop up pizza places in our area that fail).
I do like the idea of cookies - need to find a cookie lady to help us with that.