"Hand" refers to a person of nobility in service to a person of higher nobility. "The Hand of the King", a queens "hand-maidens" (not commoners but nobility).
If it referred to someone who merely worked with their hands, it your name would indicate what that work was: Carpenter, Smith, Thatcher, etc.
Your family was likely a family of medium nobility or higher nobility who had fallen on hard times losing wealth and reputation who then went on for generations earning the trust to worked as the dedicated "hand" of a high noble family. Ancestors would have had to have been highly skilled in social graces and diplomacy, intelligent, cunning even, yet loyal and honest to their reports, enough so that they were given authority to speak and act with the full authority of the person for whom they were hand. In fact, hands were often employed by those who felt less confident in their abilities to deal with issues directly.
There is also the possibility that the name means Jude, as "hand" translated to Hebrew is "יד" or Yiddish "Jd" (Hebrew rarely employs vowels), thus "Jd" to "Jud" or "Jude". In addition, Jewish organizations have been known to use the word "Hand" in naming themselves, such as the "Black Hand" referred to the Jewish Mafia in the early 20's.
By the way, in middle school and high school I knew a girl whose name was Heidi Hand.