Sneakers
There won't be a place on the island that does not offer the temptation to walk around. Take your sneakers always with you, you will need them!
Sunglasses
There is no holiday if you don't wear sunglasses. But besides looking cool, they protect your eyes from the fierce Canarian sun. Use no lower than category 3 or 4.
A jacket
Regardless the season, you will need it at some point. For your nights having dinner on a terrace or your walks in the Garajonay mist. Pick an all-terrain one!
Swimsuits
Not one, but 2 comfy swimsuits, and for multiple reasons: high humidity makes it hard for clothing to dry; because you will not want to get on the car with a soaking bottom; and comfortable one because you will at some point forget to redo your swimsuit after each wave.
All-terrain flip-flops
Ask yourself this: Can you swim with your flip-flops on without them falling off? Then your golden. If not, find you ones that will stay on your feet no matter what. Rocky natural pools or wavy rock beaches... They ain't so much fun in wobbly shoes or barefoot.
Reusable bottle
Keep your water fresh all day and avoid using plastic bottles every time you are thirsty (and it will be very often). A tip: if you see your water reserves waning, ask at any bar for a refill, they will do it gladly when asked nicely and with a smile.