I asked in Ireland and there seem to be as many ways of saying it as playing it! The most common seems to be bow-(as in now)rawn (as in torn). What you definitely don't call it is a 'bodran'

As a bodhran player who does occasional workshops (four days running once a year is quite enough believe me)I am sometimes blamed for the number of drums turning up at sessions. Unfortunately there are a lot of people out there who buy a bodhran and assume that owning one is qualification enough to bring it along and bang it. You wouldn't buy a guitar or a flute and turn up expecting to play without learning how - or would you?