How can I hang a plywood sign by the edge?
A large sign should be hung with a fastener that is passes through a hole that is drilled through the face of the plywood. 

Using fasteners into the edges of plywood is almost always to be discouraged because the threads of screws or lag bolt type fasteners just do not hold very well. In the case of a sign the hole in the top edge invites the possibility of moisture intrusion and the separation of layers which can further weaken an edge joint. 

There are various types of hanger brackets that can be used for a face drilled fastener in the sign. Here is a picture of one type that could be used: