Called "urophagia", drinking your own urine is practised for various reasons, one of which, urine therapy, is a form of alternative medicine although its medical benefits are widely disputed.
Urine is made up by a large amount of water, precious to someone stranded on a baking, arid, desert island. It is also common did-you-know knowledge that, urine is relatively bacteria free, although when giving urine samples it is sometimes requested that the sample start mid-flow so that the first couple of seconds can be used to pass any bacteria that might be clinging to the walls of the urethra. Perhaps it would be a good idea to quench your thirst with this free liquid source?
So, should you drink urine if you're stranded on a desert island? The answer is no.
Passing urine is a way for the body to get rid of soluble materials that would otherwise harm the body. Your kidneys filter your blood, concentrating sodium and other minerals into water to create urine.
If you were to drink urine the main dangers would be this high sodium and mineral content, while dehydrated urine also has a high concentration of urea all of which would prove toxic to your body. They are being expelled for the very same reason why you shouldn't ingest them!
If you were desperate, you could evaporate drinking water from your urine using a solar still or some other evaporation-condensation method, thereby separating the waste products and the precious clean water. Otherwise, maybe stay away from desert islands.