The blue leash is the object that Rusty initially used to transport Pikachu while he was still a Kakuna. When Rusty did not want to carry Pikachu, he dragged him with this leash.
The blue leash is exactly that. It is a relatively short leash that is of a light blue or baby blue color. The reason for it seeming so short may be due to the way that Rusty ties it around Pikachu or another pokemon of his.