When I was young, only the priest gave the eulogy at the funeral Mass. Allowing laity to give eulogies was a post-Vatican II innovation taken from the Protestants.
The wake, in which people gathered together at the home of the deceased, was a time to gather together to share a meal and a Rosary. Sometimes there was a eulogy or two given by a relative, but it consisted more of a sharing of memories, usually funny and/or spiritually uplifting.
I attended a novus ordo funeral recently in which the nephew of the deceased gave a welcoming address and eulogy in the mortuary chapel before the priest arrived for the funeral mass.