NavList:

Yes - I have used a stopwatch since the 1980s and still do, though not the mechanical variety.

I find the lap time function on a quartz stopwatch or wristwatch very useful as the watch can be set to the ship's "chronometer" (inevitably just another quartz watch) to provide a "start time" and then a series of sights taken.  The sight times can be recalled via the lap time function and then added to the start time. 

I have always prefered a cheap digitial stopwatch for this purpose as it reduces the chance of error when reading off the time.  With practice the stopwatch, or suitably sized quartz wristwatch can be held in the right hand holding the sextant and the lap time button depressed.  I have also used a small dictaphone recorder so that I can call off the altitudes without having to pause and write them down.  This is great for solo work so long as there is no water coming across the deck.  With a lanyard on the stopwatch and around either the neck or the wrist (there is usually a lanyard around the neck attached to the sextant) there is no concern if the watch is dropped.

Back in the days before electronic charts and GPS the stopwatch had other uses on board expecially for pilotage close inshore.  I remember sailing through a narrow channel with hidden obstructions where the point at which course changes were made was determined using a combination of pre-determined bearings on shore-based landmarks and time-on-distance measurements based on course steered, speed and stopwatch time.  Not as dramatatic as a certain scene in the film "The Hunt for Red October", but the same general idea.