With the new design of SE, the previous answer isn't exactly accurate, though the steps are almost the same.

1. Click your name at the top
2. Click on "Edit Profile & Settings"
3. Change your display name and Save.

Note, that you can only change your display on the main site. The setting isn't available if you open your profile on a meta site.

With the new design of SE, the previous answer isn't valid anymore, though the steps are almost the same.

1. Click your name at the top
2. Click on "Edit Profile & Settings"
3. Change your display name and Save.

Note, that you can only change your display on the main site. The setting isn't available if you edit your profile on a meta site.