Coptic pope

When the Coptic Pope (technically, the "Pope of Alexandria and Patriarch of all Africa on the Holy See of St. Mark the Apostle", or in full at least according to Wikipedia the "Pope and Lord Archbishop of the Great City of Alexandria and Patriarch of All Africa on the Holy Orthodox and Apostolic Throne of Saint Mark the Evangelist and Holy Apostle that is, in Egypt, Pentapolis, Libya, Nubia, Sudan, Ethiopia, Eritrea and all Africa") dies, a new pope is chosen by having a bunch of names of nominees put into a hat (not literally a hat, apparently, but at least a box) and having a child from the congregation pick a name out.  When this happened in 1971, the child was even blindfolded, to make sure the choice was "in god's hands."

Full details of the various election processes (including the current one as revised in 1957) can be found in a PDF here.