When you name your group "The Avengers" presumably you are avenging something (or someone) right?
I'm afraid they're not really avenging anything. At the end of the first issue the Hulk points out that the group needs a name and the Wasp suggests "Avengers" because it's "colorful and dramatic."
In the recent The Avengers film, they don't embrace the "Avengers" name until
Agent Coulson's death.
So, that's what they're avenging. Although Stark's line "If we can't protect the Earth, you can be damn sure we'll avenge it," makes for a nice snippet of dialogue for trailers, the spoiler is what the plot of the film has them avenging.
In the MCU we now have a clear reason for the name. In the 2019 movie Captain Marvel, we learn that
Nick Fury named his proposal for "The Avengers Initiative" on Carol Danvers' call sign, printed on her plane as Capt Carol "Avenger" Danvers
which he saw in a file photo of her plane when she'd been a test pilot for Pegasus. The link is very clear; the paper was titled "The Protectors Initiative" until Fury saw the photo, reconsidered, then went back and edited the document to reflect the Avengers name.
The name appears blanked out in this photo, which was available before the movie release, but the pilot's name would be printed where the white rectangle is: