"The big mamou" or "the grand mamou" is an American idiom meaning to pull out all the stops, bring out the big guns, and do it right. It is not very commonly heard even in America, being primarily found in and around the state of Louisiana.
The origin of 'mamou' is highly debatable. Most likely it comes from the French mammouth, meaning 'mammoth' -- in this case, specifically in the sense of 'big'. The Mammoth Prairie is a large prairie of over 1,200,000 acres in south-western Louisiana, and has been referred to as such in both English and French since the 1700s. The name lives on in the town of Mamou, LA, a small town known as "the Cajun Music Capital of the World" (n.b., however, that Opelousas is "the Zydeco Music Capital of the World").
Les Mardi Gras devient de tout partout
Mais principalement di Grand Mamou
--La Chanson des Mardi-Gras, traditional folk song.
Either origin, refering to 'mammoth' or the chance to hit it big in Mamou, would be enough to explain the term, but there are also references to Native American origins -- a local Chief Mamou or a type of bean locally referred to as the Mamou bean, which has a tall stalk, thorns, and bright red flowers. While it remains uncertain who or what had the name first, the modern usage most likely comes in reference to the town or prairie.
It is perhaps worth noting that the most prominent usage of the term "the big Mamou" in modern times appears to be as a Very Cajun way to name random menu items, from crayfish dishes to cocktails to breakfast biscuits. There are also a few restaurants of this name. In this context, Mamou is flavor text, and indicates nothing specific about the dish, other than that it is targeted at tourists.