L.A.
She needs to figure out what exactly about the elevator frightens him. Remember she needs to get on his level and look and see what he sees. She also needs to listen to the sounds and smell all odors.
Since he is close to the bottom of the door he can see through the threshold. That means he can see down the elevator shaft and that can look like a big drop. He does not have proportion understanding, so he may think he is going to fall down in there..
It is just like children who all of a sudden become afraid of the bath because of the drain.
She will need to maybe make a deal with him that she will carry him on and off.
I would take one of his empty shoes and show him there is no way for the shoe to fall in there. She could also have him hop over the threshold till he gets used to it.
The other thing is elevators do not always give a warning when the door is about to close. He may be afraid of getting hurt or stuck in the doors. Once again, your friend (may want to try this without her son), let him see what it feels like when the door closes so she can give him reassurance it will not hurt if mom holds the door.
Also have him be her big helper and push the button if no one else has pushed it yet. Good way to learn what the numbers look like.