There were some rumors floating around recently that Sony was working on a way to allow users to change their PlayStation Network Online ID. Apparently they were true, as the PlayStation Blog confirmed in a post this morning that the ability to change your ID from your PlayStation console will be entering the Preview Program soon with full rollout scheduled for 2019.
For those that are looking to get in on the action immediately, you may still be able to get access by registering to be part of the PlayStation Preview Program and pre-register as testers for previous PS4 system Betas. During testing, you can change your ID as many times as you want, with the first change free. However, any subsequent changes will cost $9.99, or $4.99 is you are a PlayStation Plus member.
Couple other caveats besides the cost. There is an option that will allow you to display your old name alongside your new name, so your friends can more easily recognize you if desired. However, while the feature is compatible with any PS4 title published after 04/01/18, and should be with most PS4 titles released prior, it was noted that not all games will support it across the PS4, PS3 and PS Vita systems.