In Chapter 6 of "To Kill A Mockingbird," what is Mr. Avery's claim to fame?
If you are talking solely about Chapter 6, then I would say that Mr. Avery's claim to fame is that he can urinate a long distance.
In Chapter 6, Jem and Dill and Scout are walking by the house where Mr. Avery boards when they see him (it's implied) urinating. He is able to make the urine hit somewhere like ten feet away from him.
Lee doesn't exactly say that Avery is urinating, but it seems quite clear given that he is responsible for a stream of water. Also, the two boys then hold a contest to see who can best copy Mr. Avery and Scout is "untalented" in that way, which implies that they are seeing who can urinate farthest, which is obviously not something a girl would be able to compete in.
To add to the previous response, the author puts this episode as a divider between Jem and Scout. Miss Maudie will act as a female role model for Scout. The fact that Dill and Jem are fascinated by Mr. Avery's talents and wish to imitate leaves Scout out. With no mother, and only secondary female figures in her life, Miss Maudie will demonstrate a more feminine image of strength to coincide with Jem's admiration of the gentlemanly stature of Atticus.