The 94-year-old, who became an icon through the anti-apartheid struggle, is currently in critical condition in a Pretoria hospital with a lung infection.
Hamilton has revealed that he first met Mandela a few years ago and has since stayed in touch.
"He has become like a grandfather to me - that's how I look at him," Hamilton wrote in his BBC Sport column. "He has had such a huge influence on everyone in the world.
"It is remarkable to meet someone like that, who has achieved so much. It makes you realise how little I have achieved in my life and how much harder I have to work, and how much more there is to do."
Meanwhile, Hamilton is preparing for this Sunday's British Grand Prix.