Lawr.

Lawr. is a clerk's shortening of the given name Lawrence. Busy writers cut common first names down to a few letters, often raising the last letters above the line, and the habit runs through deeds, wills, tax lists and census pages.

The name was spelled both Lawrence and Laurence, and clerks shortened each one, so expect Lau. as well as Lawr. Both point to the same name, and neither tells you how the family itself spelled it.