To change your display name, start off by clicking the More button in the bottom right hand corner of your screen. (See image below.)
Once either the personal trainer or the client are on the More screen, click on the category that is labeled Profile. (See image below.)
You will then be on your profile page. Right underneath Update Photo, there will be a category labeled Display Name. Simply click on this category and change it to what you want your updated display name to be. After you have successfully changed it, your data will automatically save. You are all set!