I think that there is possibly confusion here between may, can and would.
It is possible that she once used to say expressions like:
Can I have ...
Can I get ...
and was taught that it was more polite to use may rather than can in that context.
Although strictly, can relates to the ability to do something, whereas may concerns permission to do something, can is often used instead of may in constructs such as the above. That works acceptably in the first person: "Can/May I [do something]".
Can and may are also used interchangeably is expressions such as:
You may leave now.
You can leave now.
Although both are intended to give permission, again - strictly - the latter relates to the ability to leave (as if the person were previously locked in!).
So can and may are often interchanged when asking or giving permission, but, in your niece's case, she was doing neither: she was requesting someone else to do something. In that case, can still works, strictly meaning "Are you able to pass the salt?" (maybe the other person couldn't reach it!), although it is really being used to mean "Would you please pass the salt?". In this case, may is wrong because she is not asking or giving permission: she is making a request.
may and can are used interchangeably when asking or giving permission.
would (or will) and can (or could) are used interchangeably when making a request.
[I was going to support this by referring to dictionary definitions, but @terdon's answer (with definitions) got posted while I was still writing mine, so I've omitted doing that.]