If someone asks GeekyChick1984's mom what her grandchild's Jewish name is, she may be embarrassed at having to explain why the child does not have one. It sounds like she is prepared to lie if she has to (give the child a name without a ceremony).
I snipped the quote above to that short bit because I'm guessing that is exactly what is happening here.
In my experience, unless you belong to a synagogue, in other words, a paid member, even the parents of a baby can't just show up to services one fine Saturday morning and say, "please do a baby naming ceremony for us." Maybe they could go to the rabbi in advance to make specific arrangements. But the grandparent showing up with a baby without the parents present? I simply cannot imagine any rabbi honoring this kind of request. It's just too out of the ordinary. The rabbi would almost certainly need to chat with the parents before doing this kind of thing.
Anyway, as others have said, the naming doesn't do anything except, well, give the baby a name. It doesn't make the baby Jewish (as PP said, baby is considered Jewish automatically by virtue of being born to a Jewish mother) and it doesn't make the baby a member of anything. And really, the naming in the synagogue is more just a formality. Parents can grant a baby a Hebrew name simply by stating that xxx is the baby's Hebrew name. It doesn't actually have to be written on any certificate nor does an official ceremony actually have to take place.
So it sounds like the OP's mom is just going to announce that the baby's Hebrew name is xxx -- sort of to whomever will listen. She probably hopes she can somehow convince you to have the child participate in Jewish rituals later and at that time she'll be able to say the Hebrew name is xxx.
So, there's really little you can do to stop her from telling people the Hebrew name she's chosen if she actually does so. At this point, there's no harm in it except the harm she is causing by not respecting your wishes and your boundaries.
Since she lives far away, it sounds like you'll keep control over visits between your child and her. What I would be wary of is the possibility of her trying to teach her religion against your will. But since this accident which is causing her new-found religion happened so recently, it might very well fade out by the time your child is actually old enough for that sort of thing to matter.