...I've been studying yoga part-time for about 2 and half years now. I'm not an "expert", but I can tell you that yoga is an EXCELLENT choice for CFS. It's not like other exercise because it doesn't tire you out - it leaves you feeling energized. This is because (pretty much all) the poses in some way stimulate your circulation and all that stress and fatigue that's built up in your joints and muscle fibers.
I'd recommend doing a short but regular routine (i.e. 15-20 mins 3-5 times a week - ideally in the morning right after you get up) of simple sun salutations (including downward dog and forward bend). Also, if you can do headstand it's definitely a good energy one (if not just do half-headstand - put your head in your hands with your crown on the ground and just rise up on your tip-toes but don't try to go all the way).
Most importantly, whatever poses you do, it's really important to focus on the breathing - in general yes, but especially for CFS. I think this would really help you. Deep, deep breathing standing in mountain pose to start your routine, and focus on your breath all the way through. There are also some breathing exercises that you can do just sitting down that really get circulation flowing (just do a google of 'yoga Pranayama' or 'yoga breathing'). Have you tried meditation?
Also I definitely agree with Heather that corpse pose is way more relaxing than just regular lying down! Sometimes when I can't get to sleep I put myself in that position with palms up and all and it usually does this trick!
Anyway, hope some of this is helpful to you and that you're feeling better soon!