There are four places for hymns during the funeral: entrance, presentation of the gifts, communion, and recessional. In addition, there is the responsorial psalm usually led by the cantor and the hymn of farewell if your body or ashes are present for the funeral. During the presentation of the gifts and communion, you can opt to have the organist play appropriate music rather than singing a hymn.
Amazing Grace
Be not Afraid
Hail Mary, Gentle Woman
How Great Thou Art
I am the Bread of Life
In Paradisum
Irish Blessing
Joyful, Joyful, We Adore Thee
Lead Me Lord
On Eagles' Wings
One Bread, One Body
Prayer of St. Francis
Precious Lord, Take My Hand
Rain Down
Shepherd Me, O God
Sing to the Mountains
Song of Farewell
Song of Farewell (2nd version) (Tune of Londonderry Air)
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Click Here for a Listing of Other Appropriate Funeral Music