Technically, what you're looking to accomplish falls into the scope of visual effects (commonly abbreviated VFX), more than editing. Editing is mostly making decisions about timing cuts between audio and video tracks. I don't mean to be pedantic, but perhaps this distinction will help your google searches.
Basically, you want to perform two actions:
- Track the eyes
- Adjust the color of the eyes over time
For the first task, I would recommend researching "motion tracking"
For the second, I would recommend "color correction"
Adobe After Effects (with the bundled Mocha module) and Davinci Resolve are each capable of accomplishing both of these tasks. I'm sure there's more, but these are the software packages I'm most familiar with. Nuke, Houdini, Premiere, Shake, FCP7 (FCPX can't keyframe color adjustments), and Apple Motion are all other names that come to mind, but I'm less familiar with them.
edit Bukkojot correctly pointed out in the comments that I forgot to mention matting and masking, which is the process of isolating the effect on the eyes, and making sure that the effect doesn't show through occlusions, such as blinking or foreground objects passing between the camera and subject.