C19: so called because of their serrated bills
Collins English Dictionary - Complete & Unabridged 2012 Digital Edition © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012
Examples from the Web for sawbill
Saw the sawbill duck once or twice riding down on a log and fired twice, unsuccessfully.Mount Rainier
Spic′ūlum, a spicule; Spike′bill, a merganser, a sawbill: the marbled godwit.