I have been noticing more ball control offense, now that you mention it. I think I remember some team just set a (franchise?) record long 17-play drive.
Getting back to nicknames for great defenses, I started building a list. It’s nowhere near complete but I think it’s a decent start.
1940s — Monsters of the Midway (and 1980s)
1960s — Purple People Eaters
1960s — Fearsome Foursome
1960s, late — The Redwood Forest
1960s, late — Doomsday
1970s, early — No Name
1970s — Steel Curtain
1970s — Sack Pack
1970s, late — Orange Crush
1970s, late — Bruise Brothers
1970s, late — Doomsday II
1970s, late — Gold Rush
1977 — Gritz Blitz
1980s, early — New York Sack Exchange
1980s, early — Crunch Bunch
1980s, early — Silver Rush
1980s, early — Killer Bs
1980s — Big Blue Wrecking Crew
1980s — Da Bears (can refer to entire team but was mostly inspired by their great defense)
1980s — Monsters of the Midway (and above, 1940s)
1980s, late — Dome Patrol
1980s, late — Miami Pound Machine
1980s, late — SWAT team
1990s, early — Gang Green
1990s — The Border Patrol
1990s — Blitzburgh
2000s, early — Homeland Defense
2010s, early — Bulls on Parade
2010s, early — Sack Nation
2010s, early — Sons of Anarchy
2015 — No Fly Zone
2017 — Jackson 5
2017 — Sacksonville
2010s, early — The Legion of Boom
2010s, late — The Ghosts (“seeing ghosts”)
2010s, late — New Jack City
2010s, late — Purple Pain
2020s — Department of Defense
2020s — The Gravediggers
2020s, early — Blitz, Inc.
Now do offense! The only ones I can think of offhand are the hogs and the greatest show on turf.
They don’t seem to be as likely to have nicknames. A few I can think of:
- Air Coryell (early 1980s Chargers)
- Ground Chuck (early 1980s Seahawks)
- The Fun Bunch (1980s Redskins, esp. the wide receivers, also known as “The Smurfs”)
IMHO it is more likely for individuals on the offense to have nicknames than groups. While with defenses the opposite is true. I am not sure why that is.
Maybe it’s because you often have multiple people involved in tackling/sacking but just one guy scoring a touchdown or throwing a football.
Greatest Show on Turf (St Louis Rams)
Which @EllisDee had mentioned, in his post to which I was replying. 
Damnit, I was looking for capital letters
Sorry about that, I generally dictate my posts on the dope and it’s kind of a pain in the butt to go back and capitalize. I was lazy.
Lest we forget — the West Coast Offense!
LOL, that’s a style of play not a nickname. Like Air Raid. 
It must certainly is a nickname.
My buddy the Bills fan always refers to the Jim Kelly era as the “K-Gun offense.” I don’t know if that’s a nickname or just the actual name of the system, like West Coast offense.
I guess the Run and Shoot would also be in that group.
We’re talking about nicknames for a particular group of offensive players, the West Coast Offense was an offensive strategy Bill Walsh came up with. It’s not the same thing at all.
It’s like counting “Air Raid” and “Run and Shoot” as nicknames.
What players were part of the West Coast Offense?
That’s an excellent definition. If it doesn’t identify a very specific group of players, it’s not a nickname.
Like when we say Greatest Show and Turf, we immediately think Kurt Warner, Marshall Faulk, Isaac Bruce and Torry Holt. (I had to Google the receivers’ names, sadly. Memory is not what it used to be.)
When the term was first used, it was solidly associated with Joe Montana, Roger Craig, Tom Rathman, Taylor, Rice, etc. It’s since been used for any similar offense, sure.
As for more recent, more specific nicknames for offensive units, the current crop of offensive skill players for the 49ers - George Kittle, Deebo Samuel, Brandon Aiyuk, Juszczyk, Jennings, and joined this season by Christian McCaffrey - has earned “YAC Bros.” as their name. YAC meaning yards after catch, which the group leads the league in fairly handily.
As for the defense, the most obvious nickname for a 49ers defense is already used by the cheerleader squad: the Gold Rush. There plenty of other related terms, but nothing seems to stick.
It was a trick question. There is no group of players called the “West Coast Offense” because it wasn’t a group of players. It was a system, not a unit.
Yeah I guess it’s a thing.
It’s not that catchy of a nickname but it’s an impressive group for sure.
It was a system, and while it never became a nickname for any unit there were many players on the 49ers that, as groups, they became the system and the system became the methodology implemented by those groups.
This is more applicable to the 49ers than to other teams that deployed the system.