"Think with your balls" means think smart or outside the box. To think with your balls is usually a very quick and instinctive way of making decisions. It derives from the show "How Not To Live Your Life" A decision made by the balls is often a very good short term solution to a problem but will cause trouble in the future. Sometimes thinking with you balls is the complete opposite to thinking outside the box, it is so inside the box and obvious that no-one thought of it.
"What shall we do with all the orphans?"
"Sell them on, make a tidy profit."
"Thats amazing, how did you think of that! Now were making loads of money."
"Think with your balls lad!"