(from wikipedia.org)Spain at that time ruled Milan, Sicily and the kingdom of Naples. The French politician Mazarin, an Italian himself, had attempted to liberate Naples from the Spanish rule against which the locals had fought; but an expedition in 1654 had failed in this. Mazarin was now considering Christina as a possible queen for Naples. The locals wanted no Italian duke on the throne; they would prefer a French prince. In the summer of 1656 Christina set sail for Marseille and from there travelled to Paris to discuss the matter. Officially it was said that she was negotiating her alimony arrangement with the Swedish king.
On 22 September 1656, the arrangement between her and Louis XIV was ready. He would recommend Christina as queen to the Napolitans, and serve as guarantee against Spanish aggression. On the following day she left for Pesaro, where she settled down while waiting for the outcome of this. As Queen of Naples she would be financially independent of the Swedish king, and also capable of negotiating peace between France and Spain.
Mazarin however found another arrangement to ensure peace; he strengthened this with a marriage arrangement between Louis XIV and his first cousin, Maria Theresa of Spain - the wedding took place in 1660. But this was unknown to Christina, who sent different messengers to Mazarin to remind him of their plan. In the summer of 1657 she herself returned to France, officially to visit the papal city of Avignon.
So what would happen if Christina, former Queen of Sweden, had actually become Queen of Naples?