Galatians 4, Paul recounts an illness he had that affected his eyes:
"12I plead with you, brothers, become like me, for I became like you. You have done me no wrong. 13As you know, it was because of an illness that I first preached the gospel to you. 14Even though my illness was a trial to you, you did not treat me with contempt or scorn. Instead, you welcomed me as if I were an angel of God, as if I were Christ Jesus himself. 15What has happened to all your joy? I can testify that, if you could have done so, you would have torn out your eyes and given them to me. 16Have I now become your enemy by telling you the truth?"
And in Galatians 6, Paul points out that he is using his own hand to write the letter (rather than a scribe):
"See what large letters I use as I write to you with my own hand!"
He wrote large letters because he had poor eyesight. And yet he wrote the letter (or at least part of it) himself, perhaps to make it more personal, as other Christians would have known about his ailments (or at least the Galatians would, as chapter 4 indicates). So Paul's eye problem being the thorn is a good possibility.
Another possibility is poor speaking ability, some kind of speech impediment, just like Moses was thought to have had.
1 Corinthians 2:
" 1When I came to you, brothers, I did not come with eloquence or superior wisdom as I proclaimed to you the testimony about God.[a] 2For I resolved to know nothing while I was with you except Jesus Christ and him crucified. 3I came to you in weakness and fear, and with much trembling. 4My message and my preaching were not with wise and persuasive words, but with a demonstration of the Spirit's power, 5so that your faith might not rest on men's wisdom, but on God's power."
Like others have said, we just don't know what he was referring to. I bet his original audience did, though. Whether it was one of these two problems or a different problem he had, it's obvious from several of his letters that he had a rough time and a lot of suffering.