I've looked in a surname dictionary to find the origin of Tibbett, and as you probably know there are many spelling variations. This is what it said - some early examples -John Tybote1327, Stephen Tybet 1332, James Tibbett 1674. May be diminutive of Tibb, from Tibaud (variant of Theobald). Tibbe, Tybbe 1286, 1290. In the 13th century Tibbe was used as a pet name for both men and women from Isobel or Tibald (Theobald).