I am very sorry for your loss and pray that your grandmother rests in peace.
As to your question: As a general rule of thumb, public participation in the religious aspects of the Mass ordinarily should be performed by practicing Catholics. Reading Scripture, distributing Communion, and bringing up the gifts are all religious functions (the first two in fact being *ministries *and not merely “functions”).
You might point out to your cousin, if you can do so in a gentle fashion, that there are aspects of Jewish liturgies that non-Jews are not allowed to perform (reading from the Torah, opening the ark, processing with the Torah scroll), simply because they are not Jewish. Non-Jews often are welcomed at Jewish liturgies but are there as guests who do not publicly participate in the liturgy – no matter how closely related they are to Jews who are participating in various Jewish rites the non-Jews are attending.
That said, it is possible to invite non-Catholic and non-Christian family members to take part in non-religious functions at a Catholic funeral liturgy (and other such liturgies where sacramental milestones are commemorated). Examples include greeting new arrivals, helping people to find seats, providing information about the service and the facilities (e.g., timing of the service, location of the restrooms, directions to a reception), and overseeing the guestbook.