I notice that Etymonline links the name Cunningham to the English word “coney” (or rather the variant form “coning”). According to the Dictionary of American Family Names, Cunningham is in fact “a Celtic name of uncertain origin” which first appeared in 1153 as Cunegan. It is unlikely to have anything to do with rabbits, although this did not stop the Cunningham family from including a pair of them in its coat of arms.
The botanical names Cunninghamia lanceolata, Archontophoenix cunninghamiana, Casuarina cunninghamiana could inspire the family to update the family coat of arms, maybe a pair of palm trees.