What does adding randomness mean when generating a new wallet key?

Adding randomness changes the way in which Burner's private key is generated. Normally Burner generates private keys from a combination of internal entropy and a random number seeded at the time of manufacture. When you add randomness, you incorporate a third piece of entropy into the key generation process.

Whether or not you add randomness is up to you, however, in any scenario Burner's existing entropy sources are very strong strong due to the fact that they come from distinct places and entities.