Definition: Anodyne is an adjective meaning "not likely to cause offense to anyone". Less commonly, it is used as a noun to mean "something that relieves pain" (per Merriam Webster online). The second sense could relate to either physical pain or emotional pain.
Pronunciation: An o dine
Origin: According to the Online Etymology Dictionary, anodyne comes from medieval Latin anodynus "relieving pain" and Greek anodynus "free from pain".
Why use it? The first meaning is quite similar to "inoffensive" but it has both a sort of sarcastic edge to it and also a more positive sense in that something anodyne should be soothing (which relates to the second meaning). The second meaning relates to both emotional and physical pain, although most people would probably use "analgesic" or "pain reliever" for purely physical pain. (Do not walk into a drug store and ask where they keep the anodynes!)
Examples: "Paintings in hotel rooms tend to be so anodyne as to be immediately forgettable".
"Elevator music was originally thought to be anodyne, but it actually offends many people".