I used to call to do this (typically flower delivery). Online the gift is at the bottom of the princess.com page - "Give a Guest a Gift" link. You just enter some information about who you are giving something to and then it is a normal purchase.
I don't see a gift of a specialty dining package on these pages. All I see are balcony dining on that link. Closest I can see is giving someone some onboard credit, which is easy to do and better than egift cards (they will get a card in their cabin saying you gave them a gift).
The other option: Specialty dining is done now through the princess app, so you need to log onto that. Try this login and use their booking number and birthdate: https://oceanready-personalinfo-ui.prod.ocean.com/
On the next page select Dine My Way and then specialty restaurants. But at that point you need to select a specific time and restaurant, so it doesn't fit what you want to do (although your mom could change the reservation once onboard). Problem of this approach is that if she has existing reservations this process will cancel the one for that night.