Simple answer:
A facilitator can change a client's name at any time, but, it will only stick to the session record before the client first logs in. Once the client has logged in the first time, the client will need to update their name.
Where can a facilitator edit a client's name?
They can do that in the client record:
Where can a client update their own name?
They can do that by logging in, then clicking on their profile in the upper-right corner and then "My Settings". Then find and click on the pencil icon to edit their name:
Technical Answer:
A facilitator can change the first name and last name on the client record at any time. However, those changes will only flow through to the underlying user record up to the point at which the user first logs in.
After that point, the client first and last name are independent of the user first and last name. What is shown in the session record is the user's first and last name, not the client's. So if they want the name change to appear there, the user has to change it themselves.