Though rare the surname is not "non-existent" as I first thought.
"Konijn" as a surname is mentioned in various forms by Dr. F.Debrabandere in his Dictionary of Surnames in Belgium and North France.
His estimation is that as a surname it may refer to a. the profession of rabbit-breeder b. to a certain style of nibbling and chewing food.
In the Middle Ages it occurs in the area of Ypres in a form
that is particularly interesting in this context.
In 1277 the town registers mention one Hannekin Conin. (Johnnie Rabbit).
In 1392 a Jacquemin Keunin is mentioned. (James Rabbit).
The Flemish variant may show how easily Dutch Konijn can develop into a shortened Flemish form Conin that is not unlike Coney.
Speculation of course , and as said by fjohn "no one seems to know for sure".