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In an earlier post (here), I referred to them as the Lundayeh people and also suggested that those who live near the coast, in Lawas and Limbang, are called Murut. However, the owner of the Merarap Hot Springs resort insisted that only those who live in Temburung (which is part of Brunei) are called the Murut. Those in Lawas (where they are the largest group, constituting 45% of the population) and also in Limbang (where they are a rather smaller minority, at 10% of the total) all refer to themselves as Lun Bawang. He seemed to be pretty knowledgeable about local things, so I'll take his word for it.
In fact, Wikipedia (here) suggests that the term Lundayeh is more commonly used for those who live over the state border in Sabah. I don't always trust Wikipedia, but this time it concurs with what my informant told me.