Diazepam, as others have said, is a controlled drug. That means it is illegal to supply even a small amount of it. The reason it is controlled is that it is highly addictive, no one is going to get addicted from taking one dose now and again to counteract a fear of flying.
However, drugs like diazepam can in some people have a stimulating rather than a relaxing effect. I've been one of 5 people having to restrain a bloke who had been given a drug from the same family, at a small dose, as he was anxious about a medical procedure. He was spitting in peoples faces, didn't know where he was and was trying to harm himself, having been completely fine before taking the drug.
Diazepam and other drugs in the same family can also slow your breathing down. Different people are more or less sensitive to it, I've had to admit people to intensive care who have been given a small amount of diazepam (5 or 10mg) to stop a seizure and have then stopped breathing and needed to be put on a ventilator for a few hours til the drug wore off. Mixing it with alcohol (which he may well do if on a long flight that he is nervous about) makes you more likely to stop breathing.
So he is basically thinking of making his mother break the law, risking a conviction for supplying a class C drug and also meaning the GP will not be willing to prescribe her diazepam any more, then taking a drug which may be safe for him, on the other hand may risk him having a reaction that means he can't fly as he is being restrained by the airport police as he is delusional and hallucinating, or may risk him stopping breathing during the flight.
Quite apart from anything else, he will get a similar effect as a low dose of diazepam by having an alcoholic drink or 2 before getting on the plane. This is perfectly legal, he can predict the effects if he has drunk alcohol in the past, won't make him stop breathing, he can be supplied with more on the plane once the first 'dose' has worn off, and a nice cold pint or a glass of wine or a G&T is much more enjoyable than swallowing a pill!