They used the term because the boll weevil, a southern pest, could not be eliminated by pesticides – politicians therefore thought of them as a symbol of tenacity.
The were also called Dixiecrats.
The term fell out of use in the 1980s, and conservative Democrats are now mostly known as Blue Dogs.
A related term for Republicans is the “gypsy moth Republican.”
Use of “Boll Weevil Democrat” in a sentence
Reno Gazette Journal (September 22, 2015): “James Santini was elected to Congress in 1974 and soon became known as a moderate to conservative Democrat not afraid to buck his own party. He became what was known at the time as a ‘Boll Weevil’ Democrat, even supporting President Reagan’s tax cuts in 1981.”
Texas Observer (September 26, 2016): “He’d served in Congress as a ‘boll weevil’ Democrat in the mold of Phil Gramm, winning his seat in 1978 against George W. Bush — the only election Dubya was to ever lose.”