date: 2018–08–27
I have a Logitech mouse M-705. I've lost the USB dongle.
Fortunatelly, there is an app for Linux that allows you to connect the mouse to other dongle: ltunify.
git clone https://git.lekensteyn.nl/ltunify.git
cd ltunify
make install-home
There is one more, Solaar, supposedly with a GTK UI, but I couldn't get it work – the UI just didn't appear.
Update: I found another page describing how to make it run: Restart after installation :)