Changing Buttom Functions on an M510 Mouse

If the Logitech software isn't saving or isn't preserving the settings in an accessible, writable location, then the Logitech software is somehow outdated, misconfigured, or broken. Check that the most current version of the Logitech software is installed, and confirm that Logitech supports this configuration on High Sierra 10.13 and Mojave 10.14.

Based on a look at the Logitech M510 web page, macOS isn't supported. Digging deeper, it might be supported.

The Logitech LCC software does have specific requirements for Mojave 10.14, though not for earlier releases.

As for what appears to be the Logitech software for this device, either LCC 3.9.8 or LO 7.12.82 looks to be current version. Make sure you're running that.

Failing all that, you'll then want to contact Logitech support for assistance.

I wonder if there's supposed to be a Login Item or Launch item that isn't getting installed?

To find out if it's system wide or user specific, try this...

Open System Preferences>Users & Groups, unlock the lock, click on the little plus icon, make a new admin account, log out & into the new account.

Does it work in the new account?

Might see if USB Overdrive can do it...

http://www.usboverdrive.com/USBOverdrive/Devices.html

So to summarize, a not-working device was acquired from a company that seemingly can't afford to support their products at the prices that they can charge for their wares, and you're now presuming that the problem lies with a second entity—though the first bunch explicitly claims support for the macOS software configuration you're using—and are asking a third bunch—a third bunch that lacks access to this hardware—for help.

I'd start by removing what has been installed, and installing the simplest of options available from Logitech for the M510, and see if that works.

I see at least three different software packages that reference this mouse, which is unexpected. That's a lot of redundancy, and that's a lot of software to maintain and test.

If you really want the Unifying software, also try the previous 1.2.359 version, which also indicates support for High Sierra.

I'd also tend to avoid any mouse that required extensive drivers, and devices that seemingly aren't using Bluetooth or USB for the connection. I'm also skeptical around what (else) some of the vendors of add-on drivers are accessing and might be uploading, but that's a generic skepticism around how some of the vendors of cheap devices are accruing their profits.

And yes, have a look at OverDrive.

Have Sys Prefs>Security & Privacy open when try again, see if in that Pref Pane you see an option to install anyway.