Using the end of a spool
Weld the fragment to the beginning of a new spool and use it that
way.  Most are made from metal so they aren't that easy to make
at home.  Here is another answer that lists other methods to
weld filament including using heat shrink tubing.
As you mentioned, you can use it for friction welding.
Use it for pin/studs/rivets/hinges in prints.
Throw it away.  1.75mm * 1Kg is about 330m of filament, 3mm * 1Kg is
about 110m of filament.  The leftover isn't worth much when you
consider how much is on a spool.