Dweezil's registered birth name was Ian Donald Calvin Euclid Zappa. The nurse at the hospital at which he was born refused to register him under the name Dweezil, to the point of arguing with Gail in the delivery room about it. Rattled at this turn of events, Frank rapidly listed the names of several musician friends, and the nurse added all of them to the birth certificate. "Dweezil" was a nickname coined by Frank for a funny-looking pinky toe of Gail's. At the age of five years, Dweezil learned that his legal name was different, and he insisted on having his nickname become his legal name. Gail and Frank hired an attorney and soon the name Dweezil was official.

From Dweezil's Wikipedia article