The Big Mac gave its name to McDonald's 1970s McDonaldland character known as Mayor McCheese? Wait: the product-related character was the Big Mac, a character often called the Big Mac Police? This question is invalid because no standard McDonaldland character simply took the product's name in the requested way.
McDonaldland included characters whose names echoed menu items, but the relationship between products and characters varied by advertising campaign and market. The Big Mac itself was introduced in 1967 and became one of McDonald's most recognizable branded products.
A safer historical distinction is that Mayor McCheese was a separate character with a cheeseburger-shaped head, while the Big Mac was a sandwich with two beef patties, special sauce, lettuce, cheese, pickles, and onions on a sesame-seed bun.
Because the premise is not a clean canonical fact, this item should not be published as written.