The plan for the invasion of Europe evolved considerably between mid-1943 and mid-1944 as Allied resources and ambitions increased.

Initially, the Anglo-American COSSAC (Chief of Staff to the Supreme Allied Commander) staff produced a 113-page plan for a three division landing along a 30-mile (48 km) front in May 1944. An airborne division(-) with two brigades would be dropped inland to support the amphibious assault (this was later increased to two airborne divisions). With limited landing craft and transport aircraft available, only these four divisions could take part in the initial assault on “D-Day” – reinforcing divisions would have to land on D+1 and the subsequent days. Eleven divisions would be landed within the first two weeks to reinforce the beachhead. These troops would push inland to seize ground for supply dumps and airfields that could support a breakout operation. Once a foothold had been established, the assembled force of hundred mostly-American divisions would break out and head for Germany.

Around this time Normandy had been chosen as the location for the invasion – it was close to major English ports like Portsmouth, it was within range of Allied air cover and it was between the major ports of Le Havre to the east and Cherbourg to the west. After the failure of the Dieppe Raid in 1942 and the 60% casualties taken by the Canadian raiders, Allied planners had rejected a direct assault on a heavily-defended port – the plan was now to land near to harbors and supply the invasion force with landing craft and artificial Mulberry and Gooseberry harbors until the ports could be taken.

Attacking in Normandy had another benefit: it wasn’t the obvious thing to do: the Pas de Calais was much closer to England and had a major port. The Germans already expected an attack in the area – this made the subsequent Allied deception plan, Operation Fortitude, much easier to sell to the Germans.

On December 6th, 1943, General Dwight D. Eisenhower was appointed to replace COSSAC as the Commander of Supreme Headquarters, Allied Expeditionary Forces (SHAEF). The former COSSAC and the creator of the original plan, Lieutenant General Frederick Morgan stayed on as Ike’s deputy chief of staff. Around the same time, General Bernard Montgomery was appointed as commander of 21st Army Group, making him the commander of the operation’s ground troops.

Eisenhower and Montgomery changed Morgan’s plan. Monty in particular wanted to land more troops on D-Day, which meant the landing area had to be expanded. The new Operation Neptune plan had a wider landing area 50 miles wide that stretched from the Orne River estuary in the east to the base of the Cotentin Peninsula in the west.

Taking this broader front would require five (later six) infantry divisions (the American 4th, 29th, 1st, the British 50th, the Canadian 3rd, and the British 3rd), supported by various specialist troops like the “funny” tanks of the British 79th Armoured Division and the American Range Assault Group. Two (later three) airborne divisions (the American 82nd, 101st, and the British 6th) would land to protect the flanks of the landings. These assault troops would be followed by nearly 30 other armored and infantry divisions on subsequent days.

Getting these extra troops meant the landings would have to be delayed into June. More landing craft and transport aircraft had to be marshaled to carry all these troops. This would prove to be a fateful decision, as bad June weather nearly scuttled the entire operation on multiple occasions.

https://i.imgur.com/by5V6m8.gif - the Operation Neptune plan, as it actually played out June 6th-June 12th

This broader front would make it easier for Allied troops to reach Cherbourg in the west and the Le Havre in the east. It gave the landings greater redundancy in the event one the landings on one beach failed. In a sense, the 50-mile front offered a happy medium. A smaller beachhead would be unable to take many troops on the first day, which made it a compact and weak target for the inevitable German counter-attack. A larger beachhead would be over-extended and equally vulnerable to defeat in detail. Furthermore, a wide beachhead would have been much harder to supply, since there were only two Mulberry Harbors ("A" would be off Omaha and "B" would be off Gold).

https://i.imgur.com/ywFwunF.jpg - the actual campaign to expand the Normandy beachhead – note the proximity of the beaches to Cherbourg and Le Havre

TL;DR: The 50 mile beachhead was chosen as a way to get more divisions ashore on D-Day. Roughly speaking, each division needed 10 miles of beach to get ashore and some planners wanted to land 5-6 divisions on D-Day. The wider beachhead also got Allied troops closer to ports they'd need to take later in the campaign.

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