PSN ID change feature entering beta, publicly available in 2019
Your first ID change will be free of charge, allowing you to erase any embarrassing usernames you may still have. Subsequent changes will cost £7.99/$9.99, or £3.99/$4.99 if you have PlayStation Plus. Additionally, you will have the option to display your old ID so that friends can become accustomed to the change. This cannot be changed after setting it, so your old ID be forever stained onto your profile if choose to do so. If there is any issue after changing your ID, it can be reverted free of charge.
Source (EU)
Source (US)