What should a girl do to atone for the mistake of showing her friends' pictures to her family, despite her promises not to do so, when she was thirteen years old?

If what you mentioned is true, that the girl did not fulfill her promise not to show the pictures to anyone, then she sinned by showing them to her family, because fulfilling promises is a religious requirement. According to the majority of scholars, she did not commit a sin, but she did something that ought not to have been done.

As for photography itself: 1. Contemporary scholars differ on the ruling concerning still photography, and avoidance is preferable. 2. Photographing women involves risks, such as the pictures falling into the hands of a non-mahram and causing fitnah, which occurred here by showing them to her brothers, and this is a reprehensible act. 3. If the photography involved unlawful mixing between men and women, then the matter is more serious.