An old southern idiom. Could be used to identify a place that is empty, a person who is wound up, or any number of situations where the word "tight" is used. Nobody seems to know who Dick was. Apparently he had a tough time keeping his hat on his head.
"Tighter than dick's hatband" should be "tighter than a dick's hatband" and could be written as "tighter than a detective's hatband". A phrase referring to the band on their hat and has nothing to do with the male anatomy. I believe this is a Prohibition Era phrase.
A lot of people don't seem to know what dicks hatband is. It actually refers to a male penis, the hat being a condom and the the band ofcourse being the rubber band at the base of the condom.
When something is very close fitting. Also can mean when someone is a cheapskate.
Also can be a play on words. Dick is a man's name and he wears a hat that is too tight....but it can also refer to the male genitalia and the foreskin around the head.
"Man, that dress she had on was tight as dick's hat band!"
"I will never date Johnny again. He is too cheap! Tight as dick's hat band."