The detail that best supports the idea that many of the narrator's relatives have odd personalities is:
"My mother opposed the notion strongly because, she said, the old wooden bed up there was unsafe: it was wobbly and the heavy headboard would crash down on father’s head in case the bed fell, and kill him."
This statement highlights the mother's peculiar concern regarding the bed and her vivid imagination about the potential danger it posed to the father, which illustrates an unusual way of thinking that contributes to the overall notion of eccentricity among the family members.