How to ask Firefox to suggest a password when filling out registration form on the web?

Firefox never offers to generate a password

You need to right click in a password field (Firefox 70+):

Firefox 70 includes a new ‘securely generated password’ option that works in HTML password fields.

To use it, users simply right click in an empty password box on a sign-up page or register form, then select the Fill Password > Use a Securely Generated Password option.

Source Firefox 70 is Here with New Logo, Secure Password Generator + More - OMG! Ubuntu!

An alternative solution is using the free LastPass.

It will not only suggest an uncrackable password, but will also remember it for all browsers, so it is always available and secure.

EDIT: I now recommend the free Bitwarden, as the free LastPass version has become too limited.

Firefox is able to do this. Under Preferences > Privacy and Security > Logins and Passwords it can be enabled. The option is Suggest and generate strong passwords.

For further reading, this is the ticket tracking the feature in Firefox: https://bugzilla.mozilla.org/show_bug.cgi?id=1565407

The feature requires password fields to have the attribute autocomplete="new-password" for it to work (as of now).