I also generate additional passwords, but after reading my additional password over the phone to an agent that pretty clearly would have accepted (it's giberish), I've since started storing what I deem to be a reasonable answer to the question. A random movie for the 'favorite movie' question, random name for best friend, etc.

That's probably the best solution. It would be nice to have a dictionary of answers to these sorts of things (a dictionary of names, a dictionary of cities, etc) to quickly generate these in a truly random way to try and eke a little bit more entropy out of it.

That's a good point, and a great reason to try the "make pronounceable" option on my password generator.

KeePassXC comes with a Password Generator button in the toolbar and has a Passphrase option. Just max out the word count and choose your new favourite for each account.