Let us open up the context of chapters 11 and 12.
The context is Paul's defense of his apostleship. If one is able to boast according to the standard of the flesh, then he has more reason to boast than anyone else and he begins to compile a list of reasons to prove why this is true, 11:16-28. Beginning in verse 22, he says, "Are they Hebrews? So am I. Are they Israelites? So am I. Are they descendants of Abraham? So am I." These are all genealogical factors that he has in common with all other Jews. Then in verse 23, he begins to set forth a list of comparisons in which he is proven to excel above them all. "Are they servants of Christ? (I speak as if insane) I MORE SO; in far more labors, in far more imprisonments, beaten times without number, often in danger of death."
24. “Five times I received from the Jews thirty-nine lashes.”
25. “Three times I was beaten with rods, once I was stoned, three times I was shipwrecked, a night and a day I have spent in the deep.”
26. “I have been on frequent journeys, in dangers from rivers, dangers from robbers, dangers from my countrymen, dangers from the Gentiles, dangers in the city, dangers in the wilderness, dangers on the sea, dangers among false brethren;
27. “I have been in labor and hardship, through many sleepless nights, in hunger and thirst, often without food, in cold and exposure.”
28. “Apart from such external things, there is the daily pressure on me of concern for all the churches."
From 11:29 through 12:8 he stresses the fact that these are all things Paul regards as weaknesses of the flesh. These are things that are hard to endure and that he had the right to boast in the fact that he has suffered in the flesh more than all of them. In 12:6 he says that he does not wish to boast in these things, "... but I refrain from this, so that no one will credit me with more than he sees in me or hears from me."
In verse 7, he gives the paramount reason for his capacity for boasting which was "the surpassing greatness of the revelations." To keep him from boasting and exalting himself in this he was given a "thorn in the flesh." The "thorn" represents something that is external to the flesh but that is intrusive to the flesh. In spite of his petition from God to remove it, God says "My grace is sufficient for you." It is not through Paul's own power that he is able to endure these suffering that have been imposed upon his flesh (not to mention the psychological stress that accompanies these types of experiences), it was the grace of God that enabled him to endure them and to continue to preach in spite of them. The connecting statement that links all of this to gather is in verse 10 when he says, "THEREFORE." Whatever he says next is rooted in everything he has said up to this point and he connects it to the thorn that was given him. "Therefore, I am well content with weaknesses, insults, distresses, persecutions, and difficulties for Christ's sake. Why? Because "when I am weak THEN I am strong." The thorn made him weak. Grace made him strong. NOW, he is able to rejoice in his sufferings - in his thorn.
So, because of the "the surpassing greatness of the revelations," Paul was allowed to suffer all of these hardships - thorn in the flesh - in order to keep him from exalting himself. One is not so likely to be self-exalting when he is having the hide stripped from his back with a scourge or having to go hungry or floating around in the sea or having his bones broken from being beaten with rods or lying in a pit left for dead after having been stoned. God allowed these things so that Paul would learn humility in spite of the exalted status that God had granted him. Remember what God told Ananias in Acts 9:16 "I will show him how much he must suffer for My name's sake."
I know this is a very brief exegesis of this text and I certainly do not claim it to be infallible. It is possible that I have missed it but, it seems to me that Paul has been very clear in explaining the nature of the thorn that he had been given. If anyone has a better exegesis of this text than this, I am certainly open to correction.
His problem with his eyesight has been a favored speculation even among many commentators. We do know from Gal. 6:11 that Paul seems to have had some type of deficiency with his eyes but the nature of what the thorn was seems very clearly defined in the context of 2Cor. 12. Paul uses three different descriptive terms all of which refer to the same thing. He calls it a 'thorn in the flesh', a 'messenger from Satan', and 'my weaknesses'. These weaknesses are not of a singular nature. They are described as insults, distresses, persecutions, and difficulties all of which are in the plural. I do not doubt that his seeming deficiency with his eyes would certainly fall within this menagerie of weaknesses. The 'thorn in the flesh' is simply a descriptive term which he employed to describe a host of things he was called to endure for the cause of Christ. Their purpose was to keep him humble in the midst of his exalted position. It is hard to be proud and self-exalting when someone is beating the hide off of your back with a scourge or breaking your bones with rods or stoning you. These types of experiences are by their very nature, humbling.