Was it to open a second front in western Europe to further stretch thin the German military? Or was Italy's mountainous terrain that made a bigger advance unappealing? Or something else entirely?
Good links from /u/OldWorldGlory there. I'll try to more specifically answer your question.
Planning for an invasion of Northern France actually started even before the invasion of Sicily in July 1943. An invasion here (France, not Sicily!) is a fairly direct route to Germany across terrain that is at least moderately simple to negotiate. It offered a chance to liberate a large population from Nazi occupation and remove the area's extensive natural resources from Nazi control and move it into the Allied arena. At the same time the Ruhr, the main industrial of Germany, is within striking distance on Germany's western border. Finally it was close to the main supporting bases and staging area in the United Kingdom.
British Prime Minister Churchill, and others, advocated strongly for an invasion of Italy on the basis that :-
By the time Allied forces landed in Sicily it was clear that there would be no cross-channel invasion in 1943. The forces and logistics simply were not available, so planners moved onto spring 1944 as the target period. In the mean time however, as you point out, the limitations of Italy as a route into Germany quickly became apparent.
Although the Italian government had collapsed as hoped, the German army quickly filled the vacuum and in a serious of bloody defensive battles held off the Allied forces long enough that Rome was not liberated until June 4th 1944. Eleven months after the invasion of Sicily and only two days before D-Day. The issue was not only the mountainous terrain, but also that the narrow peninsula severely limited any opportunities for out flanking the Germans and the rivers tend to flow out from the central mountains to the coast. The advance thus became a succession of heavily opposed river crossings with all the advantages lying with the Germans on the high-ground.
With Allied forces stalled in front of Monte Cassino and on the beaches at Anzio it was abundantly clear by early 1944 that continuing to attack up the Italian peninsula was not the best use of Allied resources and virtually all opposition to the cross-channel invasion ceased.
Italy was used as the jumping off point for the invasion of Southern France in August 1944 and Allied troops (including my grandfather) advanced almost all the way up to the Alps. But it was a hard slog.
Primary sources Rick Atkinson's The Guns At Last Light and The Day of Battle.
While you wait for a more targeted answer, you might be interested in some threads from the FAQ section on “Operation Overlord aka D-Day":
In "Allied Planning and German Weaknesses" and "Allied Location Selection"
Marshall, allied planning and Normandy - 5 comments, over 10 months old.
Despite its small size, this thread gives a wealth of information on the production, planning, and politics enmeshed in bringing about Operation Overlord.
D-Day invasion: Why did they select those beaches for the amphibious assaults? - 11 comments, over 9 months old.
The commenters give links and short summaries of online sources and also discuss the distribution of air cover throughout Europe during the invasion.
What were the benefits and potential downsides of selecting Normandy as the invasion point of France in WWII? - 3 comments, over 5 months old.
In addition to discusses the advantages and disadvantages of the Normandy invasion, a commenter in this thread also gives a short summary of the Western Allies' military and intelligence operations that supported and allowed for Operation Overlord's success.
Why did the U.S. choose the beaches they did for D-Day? Would it not be easier to start further up north? - 14 comments, over 1 year old.
This post discusses the advantages and disadvantages that the Pas-de-Calais had compared to the Normandy beaches as an invasion site.
Why did the United States choose Normandy for the invasion of Europe? - 10 comments, over 1 year old.
The original poster proposes Spain as an alternative landing site and other posters explore the matter.
How was D-Day (or Operation Overlord) different for each of the Allied Forces? - 14 comments, over 1 year old.
This thread gives a summary for most of the beaches and then answers questions such as the origin of the term "D-Day" and the necessity of attacking all of the beaches (particularly the infamous Omaha landing).
Were D-Day planning details shared with Stalin? - 86 comments, over 1 year old.
The commenters here discuss the Tehran conference between the USSR, US, and UK that preceded Operation Overlord.
Was the invasion of Normandy at least partly about containing the Soviets? - 9 comments, over 8 months old.
The question is not fully answered, but there are quotes pulled from Stalin's telegrams to the Western Allies in reaction to the invasion of Normandy.
In "Other Questions on Operation Overlord"