Thank you for posting here, @youngeddie.
Allow me to provide you some information about deleting a customer in your account.
The way that QuickBooks Self Employed stores customer information is different from our normal QuickBooks Online or QuickBooks Desktop products. You don't have access to a customer list, rather you just choose the customer from the drop-down on your sales forms. That being said, there isn't a way to remove a customer at this moment. I see how this would be beneficial to businesses like yours and we love hearing your feedback, as it's what helps us improve and design a better product. I'll pass along this idea to our Development Team for possible implementation in the future.
Meanwhile, what you can do is to modify the customer name in the invoice.
Here's how:
You can also visit this page that you can use to stay up-to-date on all things QuickBooks: Our Blog.
Please let me know if you have additional queries. I'm just a reply away. Have a great day and stay safe!