The following page of the Spider-Man Miles Morales tutorial contains a description of the costume called the Great Responsibility Suit.
This handmade costume Miles receives from Peter during the 3rd main mission - The Parting Gift. It does not unlock any special upgrades.
The ill-fitting costume resembles the classic Spider-Man costume. Miles appears in it during his debut in the pages of the 2011 comic book "Ultimate Fallout" #4.
The name of the costume refers to Ben Parker's famous words - "With great power comes great responsibility".