I believe it is exactly what Paul said it was. A demon sent to buffet him.
2 Corinthians 12:7-8 KJV
7 And lest I should be exalted above measure through the abundance of the revelations, there was given to me a thorn in the flesh, the messenger of Satan to buffet me, lest I should be exalted above measure.
8 For this thing I besought the Lord thrice, that it might depart from me.
Paul expands on what this buffeting entailed:
2 Corinthians 12:10 KJV
10 Therefore I take pleasure in infirmities, in reproaches, in necessities, in persecutions, in distresses for Christ's sake: for when I am weak, then am I strong.
The evil spirit caused persecution and insults and resulted in weakness. These attacks humbled Paul and beat down that pride, which was the whole point of the spirit attacking him. "...messenger of Satan to buffet me, lest I should be exalted above measure."
The concept of having a thorn is the flesh is found in Judges, where the Lord tells them that the inhabitants would be a thorn in their sides if the failed to drive them out.
Judges 2:3 KJV
3 Wherefore I also said, I will not drive them out from before you, but they shall be as thorns in your sides, and their gods shall be a snare unto you.
So the answer is that the "thorn in the flesh" was a demonic spirit that raised up persecution and other troubles. To "buffet" him.
The term is used of Jesus where He was physically struck:
Matthew 26:67 KJV
67 Then did they spit in his face, and buffeted him; and others smote him with the palms of their hands,
In the previous chapter, Paul details some of these troubles:
2 Corinthians 11:16-33 KJV
16 I say again, Let no man think me a fool; if otherwise, yet as a fool receive me, that I may boast myself a little.
17 That which I speak, I speak it not after the Lord, but as it were foolishly, in this confidence of boasting.
18 Seeing that many glory after the flesh, I will glory also.
19 For ye suffer fools gladly, seeing ye yourselves are wise.
20 For ye suffer, if a man bring you into bondage, if a man devour you, if a man take of you, if a man exalt himself, if a man smite you on the face.
21 I speak as concerning reproach, as though we had been weak. Howbeit whereinsoever any is bold, (I speak foolishly,) I am bold also.
22 Are they Hebrews? so am I. Are they Israelites? so am I. Are they the seed of Abraham? so am I.
23 Are they ministers of Christ? (I speak as a fool) I am more; in labours more abundant, in stripes above measure, in prisons more frequent, in deaths oft.
24 Of the Jews five times received I forty stripes save one.
25 Thrice was I beaten with rods, once was I stoned, thrice I suffered shipwreck, a night and a day I have been in the deep;
26 In journeyings often, in perils of waters, in perils of robbers, in perils by mine own countrymen, in perils by the heathen, in perils in the city, in perils in the wilderness, in perils in the sea, in perils among false brethren;
27 In weariness and painfulness, in watchings often, in hunger and thirst, in fastings often, in cold and nakedness.
28 Beside those things that are without, that which cometh upon me daily, the care of all the churches.
29 Who is weak, and I am not weak? who is offended, and I burn not?
30 If I must needs glory, I will glory of the things which concern mine infirmities.
31 The God and Father of our Lord Jesus Christ, which is blessed for evermore, knoweth that I lie not.
32 In Damascus the governor under Aretas the king kept the city of the Damascenes with a garrison, desirous to apprehend me:
33 And through a window in a basket was I let down by the wall, and escaped his hands.
