With tmux 2.3+, the b: format modifier shows the “basename” (or “tail”) of a path.
set-option -g status-interval 5
set-option -g automatic-rename on
set-option -g automatic-rename-format '#{b:pane_current_path}'
The FORMATS section of man tmux describes other modifiers, such as #{d:} and even #{s/foo/bar/:}.
With tmux 2.2 or older, the basename shell command can be used instead.
set-option -g status-interval 5
set-option -g automatic-rename on
set-option -g automatic-rename-format '#(basename "#{pane_current_path}")'