germany / switzerland. The Name Muller is pronounced (in the states) with a short "u" sound. With the ü is pronounced like the vowel in fuel. Same as Mueller. The umlauts make the "u" vowel long with a slight "e" pronounced at the beginning of the vowel.
the word Muller in german means A heavy tool of stone or iron (usually with a flat base and a handle) that is used to grind and mix material (as grain or drugs or pigments) against a slab of stone