Having a overblown ego, inflated sense of self-importance, perhaps a narcissist. My mother, who was from Splott, Cardiff, Wales used this expression to describe a 1960s US game show host. I was about 10 years old when she made this remark. I thought perhaps it meant that the gentleman had a kidney disease, but she said it meant that he really liked himself.
My mother said, "I don't like him, he's got a big kidney!"