"Sawyer" is one of over 100 traditional occupations still used as surnames (or in many cases, first names) in English.
A "sawyer" was someone who sawed for a living -- in most cases, cutting planks lengthwise out of squared treetrunks. Often this was done with a pit saw, using a crosscut-type saw (like a lumberjack) vertically.
The trunk was laid over the pit horizontally and one sawyer worked the saw end in the pit while his partner worked the topside end of the same saw. Incredibly hard work, but our ancestors were a bit tougher than we.