Hey there, thanks for the generous sentiments and well-intentioned question.
That kind of trial isn't something I'd consider to be particularly valuable, as the whole issue of sleep quality and overall health is too complex for it to be possible to isolate one factor like this.
I should point out that at the times when I've experienced poor sleep with the phone on, I haven't been aware of it until I woke up the next day. It's been a case of waking up feeling less well rested, thinking "shit, why do I feel so terrible?", then reaching for my phone that had been sitting on the pillow next to my head and realising that it was turned on. So it is somewhat blind, without being scientific.
I certainly don't claim this experience as clear evidence that "cellphones impair sleep" – it's just one factor among several that seem to make a difference in my case.
It's this complexity that makes these kinds of illnesses so hard to understand and overcome. For me, good, refreshing sleep has been hard to achieve for many years, even with the phone turned off (though mercifully it's getting much better lately, as is my overall health and happiness).
And I know other factors make a difference too - what I ate/drank that day, what time I got to bed/sleep, what kind of exercise/activity I did that day, what my emotional state had been that day. Any one or combination of these factors on any given day could cause worse sleep than normal, so it would be very difficult to isolate the phone being on or off as a standalone factor.
Given all this, I'm perfectly open to the possibility that the phone being on or off is actually not a factor at all.
But given that (a) there is some established evidence that cellphone transmissions alter brain waves, and (b) it takes close enough to zero time/effort to turn off the phone's transmissions before going to sleep (as well as preventing interruptions from any calls or alerts that may happen during the night), I'm content to keep up the practice of keeping transmissions turned off when I sleep.
By the way, if you want a somewhat-scientifically based explanation that may help you understand what's going on with your relative, I'd recommend reading The Biology of Belief by Bruce Lipton. He has some colorful ideas and is easily dismissed by curmudgeonly skeptics, but his approach has solid scientific basis, and it's only by following the principles he recommends that I've been able to start enjoying such good health lately.
That aside, I wish you all the best for your relative's wellbeing.