I am not descended from this Andrew or Lyman Giseburt, but I believe we have a common ancestor in the Andrew Giseburt (spelled variously) who was married to Urith Ridgely in Frederick Co. MD 12-9-1784 and moved to Fayette Co. PA between 1790 and 1800. I am descended via Frederick, b. sometime between 1784 and '90 in MD, then Daniel, b. 1815-16 Fayette Co. PA. Daniel moved off to Illinois and Missouri by the 1840s, so he is not in your line.
I haven't been able to find out more about the earlier Andrew Giseburt/Giseburts. Do you have anything going back into the colonial period? (By the way, the Ridgely side and various maternal connections will take you back into the planter/slaveholder class in tidewater Maryland and then to gentry in England.)