It is alright to wear the necklace. The question of respect and
disrespect is a relevant one though. Your mother’s caution reflects the manners
that she was taught relating to the proper handling of the Qur’ān, and, as such,
to Allah’s name as it is written on paper or any other object. These manners, or
etiquette are a reflection of one’s sincerity in everyday activities. For
example, Muslims generally will be in a state of wudū whenever handling the
Qur’ān even though this is not essential. You do it because it elevates your
action to a greater level of sincerity and acceptability with Allah. The
bathroom is not a proper place to be reading or handling the Qur’ān and it would
thus be a disrespect to handle the Qur’ān in the bathroom. All these factors are
not rules for rote memorization and following, but rather, reflect the courtesy
and decorum with which a Muslim should approach matters of how to live and do
things according to high standards.
So in wearing the necklace, one could be additionally careful with it and with
the places that one carries it into because doing so reflects a greater respect
and humility for it, but, there is no legal rule to force you to do so.