I have been to 3 catholic funeral masses in the last 10 years where eulogies were given by a family member at the mass. 2 different churches were involved.
.
The time to do that, relative to the liturgy, would be after the dismissal.
For instance, I give the eulogy for both my father and my mother, after the "Go, the Mass is ended" and before the recessional hymn.
After the end of the formal mass.
And Eulogies by family or friends are discouraged.
We had one for my Dad, by my husband. 1999. It was discouraged after that and we didn’t have one for my Mom in 2006.
The thing about eulogies is they’re fine but not in the church during the Mass or right after. My brother wanted to give a eulogy for my Mom in 2006. Fine. But not in the Church which is for the funeral Mass. Have at it at the funeral parlor during the wake for at the cemetery during the funeral but not at the church during the mass.
It was ok in 1999 for my Dad’s funeral mass where my husband gave one. I guess it was new at the time in Catholic funeral masses so they allowed them. They don’t allow them now around here and that’s fine.
If people feel the need to make a speech about the deceased, they can do it elsewhere.