How to pre-populate the username field on the login page when clicking an e-mail link? I am using the Private Message module on my site; users receive a notification by e-mail when they receive a new message.

In this e-mail, there is a link to the login page.  Many users access the page via their mobile phones, so typing in their username can be a pain (they might also forget what their username is).

Is there some way to prepopulate the username via the link in the e-mail, perhaps using tokens?
You can do it using Prepopulate module. You haven't mentioned the version you are working with. It's in dev version for D7.

When you configure the email, use token in the same format given in the example to pass value of their username, which will be prepopulated when they click on your sent link.