First way:

Call their premium 0870 support number and pay £??? to be told that they will send you an email with instructions on the procedure which involves sending them a fax then proceed to wait several days for said email without success.

Second way:

Sign in to your PayPal account, go to the Contact Us - My Account Profile - Name Change page in their help system, select Business Name Change, and enter your business's new name in the form, click submit, and Bob's your uncle.

Hope this saves someone else a little bit of time.

I found out about this thanks to this very helpful page -- thank you for sharing.

The Two ways to change your business’s name on PayPal… article by Aral Balkan, unless otherwise expressly stated, is licensed under a Creative Commons Attribution-Noncommercial 2.0 UK: England License.