If clients receive the error message "Your appointment is not booked yet. Please enter your name.", despite having entered their name when booking online, the cause is likely as follows:
Your account is set to not allow clients to pay in full, and the system essentially considers paying with a gift certificate to qualify as paying in full.
To fix the problem, you would need to change the "When you only require a deposit or card on file, should clients be offered the option of paying in full?" setting at the very bottom of Setup > Client Payments to "Yes...". Please note however that in addition to being able to enter a gift certificate voucher, clients will be able to pay in full.
The error message this produces --"Your appointment is not booked yet. Please enter your name." -- is confusing, as it does not accurately describe the problem, which we've let the development team know about.