I would ring the even if you have to just ask him on the emergency line and he doesn't actually come out, ask for his advice. Depends on the relationship with your vet I guess. It could be she's has had a foreign object in and has got an ulcer which should be fairly easy to treat
Hopefully the vet on call will phone you back. Tell him of your concerns & exactly what the problem is, describe the symptoms & he will be able to tell you if he needs to come out directly or not. With all the will in the world your YO isn't a vet & is quite possibly giving you the wrong advice. Eye problems can escalate very quickly so need tending to quite quickly.
ISHL - like all your other posters I would recommend in the strongest way possible that you get the vet out immediately and DO NOT wait until Monday. As people have said, it could be just grit or dust or something in which case the vet would probably just prescribe a mild antibiotic cream/drops. But equally it could be something much worse and waiting even 2 days could make the difference between saving the eye and losing it. Your mum's call, but surely she can see the benefits? My boy (who I'd owned for 3 weeks) had to have an eye removed after a bog-standard conjunctivitis (never treated by his previous owners) turned literally overnight into an autoimmune disease that deteriorated and deteriorated over 6 months until we had to admit that we had lost the battle. This cost £4,000 which might help your mum to see it in a different light. (Thank god for insurance!)
I hope you have spoken to the vet by now . You really can't be too careful with eyes. A few years ago my dog had a watery eye, and she was blinking and pawing at it. I spoke to the vet (it was a sunday morning of course) and he said to bring her in as he never liked to ignore eye problems. I drove in thinking she maybe had a seed in her eye, half an hour later I was driving home in shock with an appointment for the next day to have her eye removed. It turned out the problem was a tumour which had obviously been there for a while but only just started causing her discomfort. She recovered amazingly from the op and never had a problem coping with only one eye..
I don't want to scare you by telling you this, just think it might persuade your Mum to let you get the vet out before Monday if necessary. And I am speaking as a Mum myself who mainly funds daughters horse!