You have certainly heard that visa procedures have been toughened. It now costs USD 50.- to have the visa delivered
upon arrival at the airport, and only one month visas are issued. (see also https://bradtethiopiaupdate.wordpress.com/2014/12/13/visa-on-arrival/)
I do not know what the normal procedure for extension is – queuing up endlessly at immigration? – but the procedure if you overstay is complicated. You first have to go to immigration to state that you overstayed, then you have to appear in front of a magistrate who will decide upon the fine, usualy 20-25 USD (though I was told that an Italian fellow who had seriously overstayed was fined 500 USD) and you will have to pay an additional 100 USD to get a one-month extension.