How are rivnuts measured and what eyebolt fits?

The size quoted is the size of the bolt that would be used with it (the same as a normal nut - so an M8 rivnut would be used with an M8 (8mm metric) bolt.

In the example in your picture, a 1/4"-20 rivnut would need a 1/4"-20 UNC bolt (1/4" diameter, 20 teeth per inch), and would need an 8.8mm/0.375" hole (most likely, 9mm or 3/8")

They are typically listed by the size of bolt that will fit in them. For example. RN14 1/4"-20 steel rivet nuts, would accept a 1/4" bolt that has 20 threads per inch.

If you look at the details of the "rivnut" (either in the details provided on the website, or by looking them up through the manufacturer), you should be able to find some other useful dimensions. "Hole Size" should be listed somewhere in the details, which is the size of the hole that has to be drilled for the rivnut to fit into. You should also find "Grip Range", which is the range of material thicknesses that the rivnut can be used in. You should also find the "Head Diameter", and the "Length" of the rivnut.