This is a very good question, and I have always wondered exactly what Jem meant by his use of the word "runner" in this context. Reading the text carefully, Dill has just appeared in the Finch house in time for lunch. Dill angers Aunt Alexandra when he claims that he has seen his Aunt Rachel drink
"... a pint for breakfast every morning--knows she drinks two glasses full. Seen her."
When Alexandra admonishes Dill for his accusation, "Jem's eyes flashed at her," and he decided it was time for them to leave. While telling his story, Dill has been
... gnawing a chicken leg.
My guess is that Dill had not finished eating his chicken and when Jem told him to "take that runner with you," he was referring to the uneaten chicken leg.