Interesting idea - haven’t seen a specific stopwatch applet from any third-party dev. Might be an interesting little applet for someone to wet their appetite writing budgie applets!
Haven’t tried these - but in the repos there is “alarm-clock-applet” which has appindicator support so it should (in theory) appear in the panel if you have the appindicator applet added to your panel. EDIT: from the changelog looks like they have dropped appindicator support so it runs as a standalone app
sudo apt install alarm-clock-applet
There is also a standalone app appropriately called “stopwatch”
sudo apt install stopwatch
even older but still in the repos is gtimer - something like stopwatch from the picture I saw.
sudo apt install gtimer