A payname is a unique name that identifies your banq account, much like a Twitter handle or a hashtag, but the payname is tied to the account and not the user. The account payname makes it easy for people to identify and send money to that account. It is shown in the transaction details of a payment, but it is private and not searchable.
Your payname can be whatever you like. You can use spaces or special characters and it can be long. You can use your name if you want, but know that it will be visible to whoever you pay or are paying. The only limitation is that it needs to be unique, as there can't be multiple accounts with the exact name.