Anybody can have a Mass celebrated for anybody, Catholic or not, unless the fmaily thinks the non-catholic would not have approved of a Mass.
That said, there are three parts to the whole Funeral Liturgy, and the family gets to pick and choose which parts they want. The family also gets to choose from a TON of readings, more in my count than any of the other rites of the Church.
If you do a search, you should find some links that explain things better.