What does '1 unit of lemon juice' mean in a grandma's drink recipe?
It is to show you the ratio of juice to water, but leave the quantity up to you.

This way you can make different amounts: for example 1 unit lemon juice to 3 units water.  You could sub in pints or liters or gallons or hogsheads or whatever was appropriate for the amount of lemonade you wanted to make, according to the ratio provided.  

The puzzler is how much sugar to add, which is dependent on the amount of fluid. My guess is that there is no sugar as this is a weight loss sort of thing, so really it will just be dilute lemon water.