It seems like you have it worked out, but for others looking for this answer, an easy way to do this is by printing to stderr. You can do that like this:
from __future__ import print_function # In python 2.7
import sys
@app.route('/button/')
def button_clicked():
print('Hello world!', file=sys.stderr)
return redirect('/')
Flask will display things printed to stderr in the console. For other ways of printing to stderr, see this stackoverflow post