I was always fascinated by the monikers or unofficial nicknames given to Divisions as well, especially the ones given to them by their enemies :
Red Devils: Unofficially given to the British Airborne by the Germans, later accepted as the official moniker of the British 1st Airborne Division. Attributed to the Red Beret they sported.
The Devil's Brigade: Unofficial designation of the 1st Special Service Force, U.S-Canadian regimental sized unit given to them by the Germans and eventually adopted by the unit.
Devils in baggy pants: Given to U.S. Airborne troopers in general and to the 82nd Airborne in particular (504th RCT specifically) again by the Germans. 1st widely used at Anzio but picked up in Normandy and later for almost any actions against U.S. airborne personnel.
The Battling Bastards of Bataan: Originally attributed to all U.S. and Philippine troops held up on the Bataan Peninsula in the defense of the Philippines. It was adopted as the unofficial moniker of the U.S. Philippine Division and the 194th Tank Battalion among miscellaneous other smaller units.
The Bloody Bucket Division: The unofficial nickname of the 28th "Keystone" Infantry Division, originally from the Pennsylvania National Guard. A nickname given to them by the Germans because of the keystone shape of their unit patch and punishing attacks on German forces but adopted unofficially by the unit especially after the Hurtgen Forest fighting, Gen Norman Cota Commanding, at Germeter, Vossenack & Schmidt along the Kall Trail where the division was losing a regiment's worth of platoon & company officers and men in fighting units every 3 to 5 days. By December '44 the battered Division was sent to a quiet sector to rest and refurbish. That sector, the Ardennes Forest, would soon become very active however and the division was all but destroyed in the opening salvos of the Battle of the Bulge. By early 1945 the shattered division had been reconstituted and was was used in the reduction of the Colmar pocket.
Guadalcanal Division: It goes without saying why the U.S. 1st Marine Division is known as the Guadalcanal Division and has that on its divisional emblem.
There are many others in all services and all armies but these readily spring to mind.