It seems to me, especially after the battle for North Africa had been won, that an invasion into Vichy France may have been the better option than one into German-occupied northern France. Vichy French troops would be less likely to resist and the local population there could still be counted on to welcome Allied troops. Furthermore, an operation here would have forced Germany to occupy all of southern France, thereby spreading their troops more thinly.
The Allies were obviously intelligently organized and had successful strategies, so there must be a good reason for why this was never done - I'm just having a hard time considering what those reasons might be.
They actually did invade the south of France. In what is known as Operation Dragoon( it was primarily and American operation)They wanted to capture the French ports of Toulon and marsellies. The reason they decided to wait long was because A) the Normandy landings were far more important, and B) the Americans wanted to wait until the Germans were thoroughly committed on other fronts and couldn't afford to defend the south of France.
The invasions were huge success and Army group G, responsible for the south of France began to do a full scale retreat back towards Germany.
To add to /u/Warband14's post on the "why not" angle, the south of France was disadvantageous logistically. Unlike the relatively narrow English Channel as a barrier, the Mediterranean provided a much wider gap that had to be overcome. This was important in a number of ways.
First, England was a great place to base fighter and bomber support for the invasion. A quick round trip meant that the planes could be heavily laden with fuel and/or ordinance and still easily return home. While fighters could be based in places like Sicily, Corsica, Italy, and North Africa to cover the south of France, the sorties in support of an invasion would have had to be a good bit longer. This would decrease what they call "loiter time", which is how long a plane can stay in the area of the battle waiting for a target to be assigned or to reveal itself before the plane has to head for home due to fuel limitations. To increase the loiter time by adding fuel, you would have had to remove weight from the aircraft for some models--likely by removing bombs. So, the plane would have had to choose between loiter time and maximum capability. Even after making that compromise, the longer trip to the invasion sites would have meant fewer sorties per plane anyway, as the minutes flying to the target and back eat up the clock.
Additionally, supporting the troops with supplies would have been more difficult. While PLUTO (Pipeline Under the Ocean) wasn't a decisive contribution to the Normandy invasions, it was illustrative of the efforts made to keep the invading troops fully supplied. Any increase in length of that supply chain would have been a difficult strain on the system. It takes time to cover any increased distance, so each ship could have supplied fewer supplies per time period merely due to the travel time involved. The relatively short jaunt across the Channel was already a big hurdle to overcome for the Allies, especially since they were fully mechanized and therefor starved for fuel. This fuel shortage was most dramatically illustrated by the Red Ball Express, a system in which trucks were used to ferry supplies during the breakout from Normandy (trucks largely driven by African-American soldiers, by the way).
On a more minor note, the increased sea journey would have been more miserable for the infantry involved and would have resulted in increased wear on equipment. The stormy Normandy crossing already affected the poor infantry that were unaccustomed to such travel. The Med is generally more peaceful than the Channel, but if you're not a sailor you can get seasick in a dead calm sea. The wear on equipment like the Mulberry harbors and specialized minesweepers and landing craft would likely have led to decreased efficiency. On an extremely minor note, the short sea journey did make it harder for the Germans to pick up the fleet as it simply sailed for a shorter amount of time.
You and /u/Warband14 rightly point out that the south of France was a vulnerable target, but there were a number of reasons to pass it up in favor of Normandy.