Darren,
That's actually not as difficult a trip as you might imagine. In the same situation, this is what I'd do.....
- Take one of the Chamonix Shuttles from your hotel to the main station in Geneva. An early morning departure would be best, but I'm not sure how early they start operating. You'd have to pre-book the shuttle.
- Travel by train from Geneva to Milano Centrale and then to Varenna. For example, there's a direct EC train departing Geneva at 07:39, arriving Milan at 11:37 (time 3H:58M). Another easy one would be a departure at 12:00, arriving Milan at 16:37 (time 4H:37M, one change in Brig).
I just researched trains to Milano Centrale. Once you're there, it's easy to connect with trains to Varenna-Esino, which is only a trip of 1H:03M. There are departures to Varenna at 12:20, 13:20, 14:20, 14:25 (2 changes), 16:20, 17:20 and 18:20 (1 change). There are others.
If you want to use a shuttle, I believe THIS is the one I used. I was very pleased with their service and would definitely use them again. There are lots of others.
If you were to rent a car, keep in mind the need for each driver to have an I.D.P. dealing with ZTL areas in Italy (you might want to read some of the recent posts on the forum on that topic) and highway tax vignettes in some countries, tolls, high fuel costs, etc. You'll also need an I.D.P. for driving in France or a certified translation of the terms of your D.L. in French.