If you have WebMail Lite installed on cPanel, you can enable a password reset feature - "Forgot password" link will be displayed on login page, upon clicking it user will be able to get password reset link to their alternate email address, if they specified such an address in their WebMail Lite account settings previously.
This feature shouldn't be confused with changing account password when user is logged in, it's designed specifically for the cases when user forgot their password. Typically it can be used alongside public signup but enabling that feature isn't required for password reset to work.
The feature is designed for use in the case of the product installed from ZIP package.
This feature relies on cPanel Integrator module and uses credentials of either your cPanel account, or root password. If you have configured password change for cPanel, you probably have those details supplied already. Otherwise, edit
data/settings/modules/CpanelIntegrator.config.json file and set "Disabled" to true there. Then in "cPanel" tab of WebMail Lite admin interface, supply cPanel host (127.0.0.1 would usually work), port number (2087 for root, 2083 for non-root user) and cPanel credentials.
You'll also need to enable StandardResetPassword module. Currently it's not included in the product package and needs to be added following these instructions. Of course, you don't have to rebuild the code directly on cPanel server, you can do that on your local webserver and then upload the changes.
Upon installing the module, be sure to run "Create/Update tables" and "Update Configuration" tools from Database Settings screen of admin interface, then in Password Reset section supply email account used for sending password reset mails. It's recommended to use SMTP options, you can either enter account details explicitly or just have those taken from WebMail Lite database.
Once that's done, users should be able to use "Forgot password" feature - but first, they need to set recovery email for that account, that's done in "Password reset" screen of the account settings:
NB: it is not enough to supply email address there; once you specify it, email message will be sent to that email address, you will need to click a link in it, and then recovery email will be set and verified.