"Murmur" means to complain in low tones, to criticize or grumble about the actions of others. It can be a subdued expression of discontent or anger (Oxford English Dictionary).
The Hebrew word usually translated into murmur has the root "to be obstinate" or "to complain, grudge." The Greek New Testament word is of uncertain origin but also has the sense of grumble.
Murmur is an interesting word in that it has no etymological root. It is one of those few words that sounds like what it is (onomatopoeia). So when you trace it to its earliest Indo-European root, the word is "mormor."
Murmur, however, does have an interesting history and usage. The word, or its variants, is used 76 times in the Standard Works but it is often translated differently in various biblical translations. For example, the Jewish Publication Society translation of Deuteronomy tells us that "you sulked (murmur) in your tents and said, 'It is because the Lord hates us that He brought us out of the land of Egypt, to hand us over to the Amorites to wipe us out."' (Deuteronomy 1:27)
Various New Testament translations also show different dimensions of murmur. The New American Bible translates Jude 1:16, "These people are complainers, disgruntled ones who live by their desires." The New International version renders the same verse "These men are grumblers and faultfinders; they follow their own evil desires."
Our favorite translation of Jude 1:16, however, is the 14th-century Wycliffe translation, which tell us, "These are grouchers, full of (com)plaints, wandering after their desires."
In modern times murmur is still used in the same sense. Elder Joseph B. Wirthlin of the Quorum of the Twelve spoke of those who "raise their right hand to sustain church leaders and they murmur and complain when a decision does not square with their way of thinking." (Ensign, November 1992)