Former AMD/ATI executive Godfrey Cheng recounts the tale of how the ATI Radeon video card series was conceived. Basically, it was Phil Eisler, currently the General Manager of NVIDIA's GeForce NOW Cloud Gaming, who came up with the name.
At the time, Eisler was the director of component marketing for ATI. Because a graphics chip emits "rays" of light, he wanted to use "Radium" as the name of the new product category. However, this idea got shot down due to the association with radiation and nuclear weapons.
Interestingly, Eisler discussed the idea with his mom, and she came up with the "eon" part, which she borrowed from the name of Raytheon, the defense company.