In the American version published by Scholastic Press in chapter eleven there is the text:
Ron and Hermione joined Neville, Seamus, and Dean the West Ham fan up in the top row. As a surprise for Harry, they had painted a large banner on one of the sheets Scabbers had ruined. It said Potter for President, and Dean, who was good at drawing, had done a large Gryffindor lion underneath. Then Hermione had performed a tricky little charm so that the paint flashed different colors.
and a few lines later:
Out of the corner of his eye he (i.e. Harry) saw the fluttering banner high above, flashing Potter for President over the crowd. His heart skipped. He felt braver.
Annoyingly my niece has my UK edition so I can't check that - can anyone else check to see what text it has?
Anyhow, I don't think there is any great significance to the phrase "Potter for President". It's just meant as encouragement for Harry, and indeed it worked.