Can be tricky. I mean, the family may have their heart set on some pieces that simply do not serve the Church's mandate to proclaim the resurrection and the comfort and joy of Christ's victory over the grave. They want the "familiar" but then that really sets a challenge before pastors when their wishes ask for pieces not in our hymnal (my former parish had a policy that no hymns not in the hymnal would be sung at weddings and funerals - it's kind of limiting, but it helps the pastor to be able to say: "Oh, I'm sorry. We can only do what's in the hymnal - that's our congregation's practice. Can we look at some of these?") or for pieces that simply have nothing to do with the hope and joy of resurrection.
My pastoral practice for good or ill in the past was always to insist that the first hymn HAD to be a hymn of the resurrection. It connects with the lighted paschal candle and the pall as a sign of our baptismal sharing in Christ's righteousness and victory. I'd have several suggestions available to choose from. I'd encourage the hymn before the sermon to be a favorite of the person who had died. If they had two strong favorites, maybe include another one right after the sermon.
But leaving the Church, the procession out headed to the grave? This is the time for the Church's huge "You have not won, Death! You have not won at all!" to be proclaimed. There are a handful of hymns in our hymnal that do the job, but I've come to believe that the absolute best piece at that moment is: "God's Own Child."
"There is nothing worth comparing to this life-long comfort sure. Open-eyed my grave is staring, even there I'll sleep secure. Though my flesh awaits its raising, Still my soul continues praising. I am baptized into Christ! I'm a child of paradise."
When people leave church singing THAT, they can virtually DANCE to the grave in the joy that is the unique experience of the Christian faith. Such funerals then witness as no other service of the Church does (because of the larger number of unchurched who end up showing up for funerals) of the Life that has been given the deceased, a life that is way stronger than the death that seems (but only seems!) to have triumphed. We know it is not so and sing our joy in the face of death.
Pastors, whatever you do when it comes to addressing the problem of hymns in the funeral, work to make sure that every funeral that comes along witnesses in song the joy of the resurrection itself - that the flesh that lies in that coffin dead and decaying will rise incorruptible and shining in glory!