1. Easy. Unlike for the weddings, usually we don't have much time to prepare the music to play for a funeral. We may get a call only a day in advance. Don't play anything that you can't play fluently after practicing for an hour or so.
2. Slow. On such occasion, you don't want to draw the attention of the people in attendance to yourself as being a virtuoso. Remember, your audience is deep in thoughts and prayers today.
3. Soft. Similarly to the previous point, if the organ plays too loud, then people will start to get an impression of the organ as being a central reason of their gathering. Today is not this day. Help their prayers and don't disturb them.
4. Hopeful. I personally found funeral marches of the 19th century composers, like Guilmant, Lefebure-Wely and others too depressing for the present times. I have played them, just as an experiment, but I believe people of today don't want to hear the music that is too sad - it just makes their state harder than it already is.
Having the above points in mind, here are some pieces to get you started: Bach's chorale harmonizations, some of Pachelbel's chorale preludes and some of the slower fugues and ricercares as well as some of the ornamented chorale preludes by Buxtehude. Not all of the pieces in these collections will work for funerals so choose carefully.