What did God mean when He told Paul, "My grace is sufficient for you?"

The word “grace” is found in the Bible 159 times.  The specific instance you mention is found in the very famous passage in 1 Corinthians 12:9To another faith by the same Spirit; to another the gifts of healing by the same Spirit;
See All....  Here we find Paul, a man mightily used by God to found the New Testament church, is plagued by a physical ailment (thorn in the flesh) that will not go away.   Paul says that he besought the Lord thrice to remove this affliction, but God refused.  I have a condition that helps me identify with this somewhat.  I had a very serious car accident 14 months before I was saved that has left me with a chronic neck condition.   For almost 16 years now I have dealt with this.  It has been worse at times than others.  Overall I actually do quite well and it is more of a discomfort than anything. For a long time I prayed to be healed, but never was.  I have received considerable relief over the years through exercise, stretching, proper posture and limited medical treatments, but I still have the ailment.  God has seen fit to leave me with it, but why?  Well, you see, without that accident and injury I may never have came to know God.  I was so arrogant and self centered that I had no room for Him or His people in my life.  This thorn in the flesh helped me to realize I was mortal.  It helped me to realize I was not invincible.  It humbled me in a way that only a physical injury could.  God used this physical ailment to heal me spiritually.  Today I have a wonderful wife of 15 years, we have 3 beautiful children, and God is using me in the ministry of creation science.  My life is blessed beyond measure.  Much of this is a direct result of the thorn in the flesh God gave me.  When you put this on the scales you will find that I reaped an amazing blessing from this event and condition far above the cost.  Now, looking back, I wouldn’t trade it for anything in this world.  This is what God was teaching Paul.  It is by His grace that we draw every breath.  When we put our confidence in this flesh alone we are destined for failure for it is corrupt.  God wants us to trust in Him.  By definition grace is “unmerited divine assistance given humans for their regeneration or sanctification.” Consider this thorn in the flesh divine assistance.  Paul did.  Read the passage:

2 Corinthians 12:7And 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.
See All... 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.

2 Corinthians 12:8For this thing I besought the Lord thrice, that it might depart from me.
See All... For this thing I besought the Lord thrice, that it might depart from me.

2 Corinthians 12:9And he said unto me, My grace is sufficient for thee: for my strength is made perfect in weakness. Most gladly therefore will I rather glory in my infirmities, that the power of Christ may rest upon me.
See All... And he said unto me, My grace is sufficient for thee: for my strength is made perfect in weakness. Most gladly therefore will I rather glory in my infirmities, that the power of Christ may rest upon me.

2 Corinthians 12:10Therefore 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.
See All... 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.

Wow, what an amazing revelation.  To keep Paul from being swallowed up in pride by all that God had given him by way of revelation, Paul was given a physical ailment.  Did Paul understand why?  I think so.  Look at verses 9 and 10. Rather than trusting in his flesh Paul learned to glory in his fleshly weaknesses, “for when I am weak, then am I strong.”