Lincoln was patient with McClellan because McClellan had changed a collection of ragged militia units into a formidable fighting force during the winter of 1861/62. His Peninsular Campaign came closer than anybody to taking Richmond until Grant in 1865. It was McClellan's misfortune that Lee took over for Johnston when he did, because a methodical siege of Richmond would have been right up McClellan's alley, while a free-wheeling multi-day fight was not. McClellan was more of an engineer than a bucaneer.

Eventually, Lincoln realized he needed someone who would take and keep the initiative, and went through generals until he found one who could.

I think it is actually quite unfair to McClellan to call him incompetent. He was cautious, yes. But the only thing that could have lost the war for the north was losing his army. So being cautious with it was actually a very reasonable strategy.

He was an excellent organiser and motivator of men. He also recognised the power of defensive positions and methodical advancement with modern firearms many years before WWI would make everybody else realise that. Moving to entrenched positions that forced his opponent to attack him was actually a pretty forward thinking strategy for the time. Had Lincoln and Staunton supported him better in the peninsular campaign, he might well have won the whole war right there (Grant didn't exactly do it easily, much later, with far better material advantages)

And people somehow forget that he WON the battle of Antietam against the much vaunted Lee. And that Lee named him as the best Union general he faced.

So I think the premise that McClellan was incompetent is flawed. And Lincoln put up with his arrogance because McClellan was the best option he had.

I think the most important reason was that McClellan was adored by the troops and his presence was regarded as a tremendous boost for morale. McClellan's fame and reputation brought him instant credibility among the men and for whatever faults he possessed as a tactician he was still a very charismatic, larger than life figure. His mere presence was enough to cause men to "dance and frolic like schoolboys" and could turn the mood from one of "extreme sadness" to a "delirium of delight". When Stanton and Chase criticized Lincoln for retaining McClellan after the disaster at Second Monasses Lincoln justified his decision by saying that "he had the Army with him."

Perhaps equally important was McClellan's skill at making exemplary soldiers out of what he was given. As Lincoln explained to Stanton and Chase "There is no man in the army who can lick these troops into shape half as well as he can." And perhaps more pointedly Lincoln noted that even "If he can't fight himself, he excels in making others ready to fight." McClellan was the man viewed as responsible for molding the Army into the revered professional fighting force it had become.

There were also political considerations for keeping McClellan in command. Like many of the troops under his command McClellan was no particular fan of emancipation. By removing him from command Lincoln risked further alienating soldiers already suspicious of the Republican goal of an antislavery war and risked angering Democrats who viewed McClellan as their future Presidential candidate. Removing McClellan would have been very inopportune with mid-term elections looming in 1862, for example. Appeasing the War Democrats was of utmost importance to Lincoln, particularly as they grew to distrust him on matters of emancipation.

Lastly I'm not entirely sure it's fair to characterize McClellan as incompetent. He was certainly overly cautious and had a delusionally inflated sense of self worth but he was also largely responsible for making the Union army respectable. He was clearly not the man for the job but his contributions to the war effort are often overlooked perhaps somewhat unfairly.

I'd like to add to the organizational and strategic arguments already put forth. There were off-the-field consequences for the removal of McClellan that were considered. McClellan was well-liked by his troops. Somewhat as a byproduct of his seeming reluctance to fight, they believed he truly valued their lives and would not risk them in a slaughter. The average solider was not well-disposed to his departure, though not enough to vote in a majority for him in the 1864 presidential election.

Politically, McClellan was a pro-war Democrat, a group from which Lincoln needed as much support as possible. This led to the appointment and retention of a number of Democratic generals, such as Nathaniel Banks, Benjamin Butler, and Daniel Sickles. There was political pressure to keep McClellan on just as there was pressure to get rid of him. Interestingly, McClellan was relieved of command of the Army of the Potomac on November 5, 1862, one day after the fall gubernatorial and congressional elections ended.