I used to think I had "weak" ankles. I badly sprained an ankle hiking - badly, as in it took a full year to heal- and when I saw the ankle doc he said that weak ankles are a myth. That what happens is that our habit of walking on flat surfaces has created a lazy link between the nerves of the brain and the ankle. So your ankle starts to roll- and your brain sends out a signal to the ankle to stiffen up- but the signal doesn't get there in time and bam, you get a sprain. The cure, he told me, was balance exercises. He gave me a couple to do.
Exercise 1- balance on one leg, hold the foot of the other leg out in front of you, and write the alphabet in lower case and then upper case letters, without putting your foot down on the ground.
Exercise 2- balance on one leg, with the other leg tap in front of you, to the right side, behind you, to the left side, over and over.
Yoga balance poses such as the tree pose are also fantastic for building the brain/ankle connection.
If you find any of these balance poses easy, do them with your eyes closed. It's amazing how much harder they are when you don't have a visual reference.