Scoring with "Triumphal Arch" and only one Action card in the deck
Most importantly, have fun playing with your son!

For your question, in case there is no second most common Action card in a player's deck, that number would be 0. Or maybe it's non-existent, but you certainly don't count the most common one (that's kind of the strategic point of the card). So your son should have gotten 0 VP from Triumphal Arch in that game.

Another way to think about this conceceptually is that at the end of the game, you write down how many copies you have of each Action card. You then omit then names of the Action cards (so you only have the numbers left), then order those numbers, starting with the highest. Whatever number is in the second spot after the ordering is the number you multiply with 3 to get your VP from Triumphal Arc.

The German translation is wrong in this case. I was confused as well first (not having seen the English text before), thinking you could increase the VP by deliberately producing a tie (say 4x most common Action card, 4x second most common Action).