Hi
What you need to do is:
1) Create a user profile field called 'dog_name' (or whatever you like). To do this, navigate to Administration > Site administration > Users > Accounts > User profile fields (for more info see
https://docs.moodle.org/36/en/User_profile_fields)
2) You can then get your user to add the dog's name by going to their 'profile' (top right) and then click 'edit profile'. Scroll down to 'other fields' and that is where they can enter the dog's name.
3) Add a 'custom certificate' activity to your course.
4) Click on 'restrict access' , 'add restriction', 'user profile'. Under 'user profile field' select your new field 'dog_name' (or whatever you called it). Change 'is equal to' to 'is not empty' (so that the certificate is only issued if a dog's name has been input by you or the user).
5) Edit your certificate. Select 'user field' and then click 'add element', then select 'dog_name' from 'user field' on the next screen. This will put the field with the dog's name in the certificate.
I think that's it. I hope that you have figured it all out, but post again if you get stuck