I used to do strength training 5-10 years ago. I've had M.E. for about 27 years. I'd be very careful if i was considering doing it now.
Things I learnt.
Be very careful.
Don't allow lifting to interfere, even indirectly, with anything else - literally everything else is more important, even vital when you have M.E. (as anything which wasn't probably got abandoned years ago). Just little things like, I've trained, I've done well, I'm too knackered to cook, I'll order food in - bad...means you've displaced a healthy activity with one that's not, at least at this time, for someone with M.E.
Believe me, it sneaks up on you, training feels good, at least at the time, the pain is a better pain than I felt all the time, different is better right? If you've set a goal and achieved it, plus you have different pain from normal, and lots of handy endorphins to make you feel good, that's got to be better than real life where you're stuck, in pain, unable to achieve virtually anything and certainly not anything you actually want to achieve, just stuff you have to, if you're lucky. Got to be a good thing, right? So you'll make excuses for the impact it has on other things, that you're too weak to sit down (as opposed to fall into a chair), that this will help - it may well do, it did for me, but everything else goes by degree, until you can lift more than you thought possible, but you don't have the energy to do anything else, bathe, eat, answer the phone etc.
Over the 3 years I was lifting I spent the first 6 months not able to train with more than an empty bar, I didn't start with an empty bar but eventually the realisation hits, after the 2nd or 3rd serious crash, that maybe, just maybe, using slightly less weight than seems so easy it's pointless, would be a good idea. In the next 6 months on the big 3 lifts I slowly moved up to a little under half body weight without significant issues apart from that my rest periods (between lifts) was several times longer than they would be for a pwME. In the next year I moved up to body weight on most lifts - this was hard, I'd have months where I got weaker with every session, everything else in my life was displaced by one 2 hour lifting session a week, I wasn't eating or at least not eating anything that could be classified as even vaguely healthy, if it took longer than 10 minutes to make it didn't happen, I wasn't going out, I wasn't talking to anyone or answering even official letters. Then I started having accidents, fortunately I have a safety cage which saved my bacon a few times and definitely stopped my cat from being a red smear on the floor on one occasion. I did lifts without warming up fully (the warm ups for higher weight lifts start to become aerobic exercise after a certain point), or got distracted, losing tension at the top of a 160kg deadlift is no joke, or so my lower back told me for a few months.
Think you'd never take it that far? Neither did I.
The downside (if the above doesn't apply to you - yet), over the 3 years my walking range gradually reduced. When I started, at my own pace, I could walk about 4 miles (took me all day but I could do it, just not on consecutive days). Training initially made this easier and faster, as you might expect, but more muscle means more energy burnt to achieve the same thing, if you're energy is fixed, and being burnt faster than you're used to it's perfectly possible to simply run out, 100m from home, and spend the next 3 hours trying to get the rest of the way - not fun. Muscle cells come in 2 basic flavours, those that do the strength thing and those that do the endurance thing, if you train them for strength then endurance suffers, at least I suspect this is why my walking range dropped from 4 miles unsupported to around 20m unsupported (about 100m supported) - yep, I now use crutches to go anywhere, and not because of the accident with my back. Of course, having M.E. I can't do the aerobic type exercises required to train the muscle cells back to endurance mode, so I can't fix it.
So...what am I saying?
Weight training, I'd suggest not doing body building type sets as these are aerobic, instead do low reps, low numbers of sets, at least 1 graduation down from what you can comfortably complete a set at. I'd also suggest that unless you have a specific need for a particular muscle then don't train it specifically, so no curls! As such I'd suggest there is no real point in doing exercises which don't train the major muscles all at once - instead do exercises such as squats, some form of press, and of course picking the bar up off the floor.
You can probably combine the benefits of most of these in simply placing an empty bar on the floor, and then lifting it above your head, and repeat. Bear in mind lifting weights above your head without a cage is iffy safety wise, and can (did to me) cause a crash on it's own. Only if you need to progress beyond this is there probably a point in splitting the motion up into separate exercises, or modifying the action into rows or something else.
Almost all of the benefits of training for me came in the early stages, before I started using enough weight to built muscle, by nervous system adaptation to allow it to use what muscle I had more effectively. Only after this point would you need to build more muscle to increase strength and IMO going beyond this is not necessary for most people, whether they have M.E. or not - very few lions to wrestle out there these days.
It's weightlifting, lift a weight, put it down, don't get hurt, job done - do something else that actually contributes to your survival.
FYI - I'm probably going into a crash right now, I'm on morphine to help with pain, my brains fried, my body isn't working very well and my eyes, argh, my eyes - I'm also aspi, so if the above makes sense, all good, if it's total gibberish or if I've offended anyone, been insensitive or just been too full of **** then I apologise - just tryign to help.