The roll represents assurance:
Assurance is a Protestant Christian doctrine that states that the inner witness of the Holy Spirit allows the justified disciple to know that he or she is saved. Based on the writings of St. Augustine of Hippo, assurance was historically a very important doctrine in Lutheranism and Calvinism, and remains a distinguishing doctrine of Wesleyanism and Methodism.
Wikipedia: Assurance (theology)
We need to contrast Christian, who gains his assurance immediately after conversion, signified by entering the Wicket Gate (repentance) and arriving at the foot of the Cross (the coming of faith); and Ignorance, who has done neither of these things, but simply hopped over a wall and joined the other Pilgrims on the King's Highway. Christian has assurance, but Ignorance does not, and he does not have an evidence of assurance (i.e. his own Roll).
Rather, Ignorance has what is termed false assurance. He pretends that he is a Christian, and may indeed believe himself to be a Christian, when in fact he is not, since he has come neither to faith nor repentance. He is a sham, even though he talks like a Christian, and likes to be in the company of Christians.
At the very end of the story, both Christian and Ignorance make it across the River of Death, and to the gates of the Celestial City. However, Christian is admitted, since he has his Roll, whilst Ignorance is not. Ironically, Christian has great difficulty getting across the River, whilst Ignorance gets across easily, and this irony was deliberate on Bunyan's part.
Now while I was gazing upon all these things, I turned my head to look back, and saw Ignorance come up to the river side; but he soon got over, and that without half that difficulty which the other two men met with. For it happened that there was then in that place, one Vain-hope, a ferryman, that with his boat helped him over; so he, as the other I saw, did ascend the hill, to come up to the gate, only he came alone; neither did any man meet him with the least encouragement. When he was come up to the gate, he looked up to the writing that was above, and then began to knock, supposing that entrance should have been quickly administered to him; but he was asked by the men that looked over the top of the gate, Whence came you, and what would you have? He answered, I have eat and drank in the presence of the King, and he has taught in our streets. Then they asked him for his certificate, that they might go in and show it to the King; so he fumbled in his bosom for one, and found none. Then said they, Have you none? But the man answered never a word. So they told the King, but he would not come down to see him, but commanded the two Shining Ones that conducted Christian and Hopeful to the City, to go out and take Ignorance, and bind him hand and foot, and have him away. Then they took him up, and carried him through the air to the door that I saw in the side of the hill, and put him in there. Then I saw that there was a way to hell, even from the gates of heaven, as well as from the City of Destruction. So I awoke, and behold it was a dream.
Back to reality (tl:dr)
The doctrine of assurance is not exactly controversial, since a Christian definitely needs some measure of assurance to in order to prosper and be useful in his faith, and Paul writes about full assurance in his letters. Assurance is basically confidence, and without confidence, no one can achieve anything of importance. However, some denominations major on it, whilst others simply acknowledge its desirability.
Believers may gain assurance immediately upon conversion (as does Christian), whilst others come to assurance slowly, and some not at all. Many believers lose their assurance at some point in their lives (again, like Christian), and may only regain it after a long time and a lot of struggle, and then only partially. A few manage to keep their assurance throughout their lives (I know of at least one such person), and they tend to be the most cheerful of Christians.
Bunyan, with his little Roll, would, on the face of it, have us believe that one needs assurance to enter the Kingdom of Heaven, but this is not so, as Jesus makes quite clear in the Parable of the Sheep and the Goats in Matthew 25 (my apologies for the long quote):
When the Son of man shall come in his glory, and all the holy angels with him, then shall he sit upon the throne of his glory: And before him shall be gathered all nations: and he shall separate them one from another, as a shepherd divideth his sheep from the goats: And he shall set the sheep on his right hand, but the goats on the left.
Then shall the King say unto them on his right hand, Come, ye blessed of my Father, inherit the kingdom prepared for you from the foundation of the world: For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in: Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me. Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee? or thirsty, and gave thee drink? When saw we thee a stranger, and took thee in? or naked, and clothed thee? Or when saw we thee sick, or in prison, and came unto thee? And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me.
Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels: For I was an hungred, and ye gave me no meat: I was thirsty, and ye gave me no drink: I was a stranger, and ye took me not in: naked, and ye clothed me not: sick, and in prison, and ye visited me not. Then shall they also answer him, saying, Lord, when saw we thee an hungred, or athirst, or a stranger, or naked, or sick, or in prison, and did not minister unto thee? Then shall he answer them, saying, Verily I say unto you, Inasmuch as ye did it not to one of the least of these, ye did it not to me. And these shall go away into everlasting punishment: but the righteous into life eternal.
Matthew 25: 31-46 (King James Version)
Here, the righteous have absolutely no assurance that they will be accepted, when in fact they are welcomed with open arms; whilst the unrighteous, full of false assurance and excuses, are sent away empty-handed.
For an eminent example of someone who lost their assurance, and then (hopefully) found it again in some measure (King Solomon), read Ecclesiastes: "Vanity of vanities, all is vanity."
It is possible that W. B. Yeats alludes to assurance (or, rather, the absence of it) in The Second Coming:
The best lack all conviction, while the worst
Are full of passionate intensity.
And the rough beast may be the outworking of false assurance: the behaviour of those who think that they believe in the divine, when in fact they believe only in themselves. You have only to look at Iraq and Syria to see the rough beast in action.