Well my wife plays at funerals from time to time, and recently every one has used Cwm Rhondda. One has had The Day thou Gavest, one has had Calon Lan (which can be used for several hymns, including What a Friend we have in Jesus), one Abide with Me, one The Lord's my Shepherd, one Thine be the Glory (tune by Handel), one I think Be Thou my Vision (whose tune is Irish). One had I watch the Sunrise, which is fairly modern. Here is a UK based site which gives a fuller list.

However most funerals, and all of those my wife has played at, have been for elderly people. Whenever I have been at memorial services for children (fortunately rare), they tend to chose modern hymns, e.g. Graham Kendrick ones.