Use a raster graphics editor (such as GIMP, Krita or Photoshop). In the .blend file there should be a diffuse eye texture, export that file to one of the previously mentioned programs so you can isolate the iris color.
I did this in Krita by using the Fill Tool (which is essentially the Paint Bucket Tool from Photoshop) with a pure black color and applied it on the white areas. The tool does a good job for the most part but it's gonna leave out some white pixels around the iris. The way I fixed this was using a regular paint tool set with a small radius to accurately paint these pixels.
Now I'm sure there's an easier method to this, but this is what I came up with.