I have attempted to compile here the information that was gathered by Milton Sharon and others during the past 30 years or so about the family that became the Sharons in the US, and variants in other parts of the world. A detailed index of about 700 family members is available here, along with other family history information.
Milton Sharon spelled the original Lithuanian name Charondzickui, and others have told me it was Choronzitsky and other spellings. The pronunciation was something like har-on-zit-sky, with the accent on the 3rd syllable.
Some members of the family came to the US from the village of Lazjidai, Lithuania (in Yiddish, Lazdei, pronounced "lozday", ) in the early 20th century, settling in and around Cincinnati, Ohio. Lazjidai is located in the western part of Lithuania, very near the Polish border. According to family stories, the family name Charondzickui comes from a nearby town, Charondzetz, where the family lived previously. Another part of the family had lived in the town of Vigran, from which they took the surname Vigransky or Wigransky.
Dov Aryeh Charondickui became Ben Louis Charon (pronounced like "sharon") when he arrived in the US. His brother Joseph Charon's son Jack was a dentist. They decided it would be better to spell the name with an "S" to make it easier for patients to find him in the telephone directory. Ben Sharon decided to do the same.