A friend of mine made the claim that I should empty my water kettle and not reboil the water that was left over. He said it has negative health effects but could not explain why.
Is he right, what would the reasoning be? What would these negative health effects be?
In summary, there is no risk to boiling the same water more than once. Nothing about the water changes except the temperature. There could be theoretical risks if you increased the concentration of contaminants in the water (since some of the water is lost as steam, the concentration of anything dissolved increases), however simply reboiling previously safe water in the volume contained in a kettle once isn't going to do anything.
There could also be issues if you were boiling water in something unsafe such that metals from the container are slowly dissolving in the water over time - in that case the issue is just that you are using the wrong container rather than any actual problem with reboiling.
Personally, I live where tap water is high-calcium and high-iron; both can cause some moderate issues with taste of water or resulting coffee/tea and leave some deposits behind far before the water itself becomes dangerous. For that reason, I prefer to boil water just once and not fill a kettle with too much more than I plan to use. Not for health reasons.