Sources connected tell TMZ ... the Migos rapper was the one who came up with the name Wave.
Once Cardi and Offset knew they were having a boy, we're told they started throwing around words and names for their first son ... and when Offset suggested Wave, both parents loved it.
The middle name is a much different story.
Our sources say Cardi and Offset spent much longer figuring that out ... originally toying with Marley ... a play on Cardi's middle name, Marlenis.
We're told Cardi didn't like Wave Marley Cephus -- it just didn't have the right ring -- and when the kid was born Sept. 4 and his parents saw him for the first time, Cardi and Offset decided to go with Set.
So, both names are courtesy of Offset. Most of Offset's friends call him Set.
As we reported ... Cardi B and Offset waited nearly 8 months to reveal the boy's name andpost the first pic.
We're told Cardi and Offset took their time because they get a ton of negativity on social media, especially toward their kids. They waited for the perfect occasion -- their Essence cover shoot.
Wave Set is an awesome name ... and the kid’s already ice-cold because his parents hit up celeb jeweler, Elliot Eliantte, and got him 2 Diamond studs, 3 tennis chains and a custom wave charm with a Miami link chain worth nearly $200k.