Thing can be seen throughout the series as both a right and a left hand. Ted Cassidy (and the other actors who played Thing) alternated their hands as a joke.
The wire that moves Cleopatra is visible occasionally.
Fester is sometimes referred to as being an Addams, but he is from Morticia's family, named Frump.
The pet Kitty Cat is a male lion with a massive mane but several times is referred to by the main characters as "she".
Wednesday usually says her headless doll is named Marie Antoinette but occasionally calls it Anne Boleyn instead.
When Morticia speaks French to drive Gomez wild, she often pronounces the words very badly.