MOBILE: Go to your main screen (the one with all your spaces listed) and click the "Settings" icon on the blue task bar at the top of the screen (next to where it says "Your Spaces"). Tap "Edit your Profile" and you can change your First and Last Name in the editable text boxes provided.
WEB: In the "Your Spaces" panel (where it lists all of your spaces) click the "Settings" gear wheel icon in the top left-hand corner. Click "Edit Profile" from the drop-down. If you click directly on your First and/or Last Name listed you should be able to edit the text directly.
By "Your Spaces" do you mean instead "Your Groups?"
Under "Your Groups" there is an option for "Edit Profile," and the name can be directly clicked and edited. HOWEVER, there is not way that I can see to maintain the change. As soon as one clicks outside of the name edit field, or hits "Return," the original name returns.
@Julie Bethman just wanted to point out what I found out myself first hand. when you change your profile name etc, make sure you press "Enter" or "Return" as this was the final step @Brianna neglected to mention.
Again, edit your screen name, then press the enter or return key. A message should display informing you that your new name has been changed etc.