In the customer area you have the possibility to send an account invitation to an existing customer who does not yet have an account.
To be able to send the account invitation, go to your shop administration under "Customer > Customers" and select the right customer from the list.
Then click on the button "Options > Send account invitation". Now he will receive all further necessary steps by e-mail.
However, it is important that the customer has already ordered or you have already created the customer so that he appears in the customer overview. If you want to know how to create a customer, you will find a tutorial here: Creating a Customer