The key word within this question is “allow”. God does not “make” suffering or create suffering himself, but He does allow it for specific reasons. We need to remember that suffering does not come from God; it is not His desire. Suffering, and all of evil, is not created. Suffering is a twisting or corruption of creation. Suffering is not God’s creation; it is a result of the Fall of man. God allows suffering because He allows the Fall (He allows man to have a free will). God is not the source of evil/suffering. Satan is the father of evil (Jn 8:44). Sin is the mother of evil (James 1:15). Suffering is part of the Fall of man and the fallen world. When truth is twisted, it can “break” and the process of breaking is painful; it is not the way it was meant to be. When God’s righteousness is corrupted, it becomes “wrongness” instead of “rightness” and that causes suffering because the corruption puts things “out of whack”.

Another key theological concept related to this question is that God is sovereign. God did not create suffering but He remains in control of it. God’s control is absolute and, thus, He uses suffering. God uses suffering in the same way that He uses all things; that is, for His glory (Jn 11:14). He can use suffering for the glory He gains from using it to sanctify us. He can use suffering for the glory He gains from using it to reveal Himself to others. He can use suffering for the glory He gains from using it to bring forth good; that is, good comes out of evil (Rom 8:28). He can use suffering for the glory He gains from using it as a tutor (light becomes more obvious when put up against darkness); that is, to point man to the horrors of himself and to the grandeur of God (via discipline, warning, instruction, etc...). God is not to blame for suffering but He does allow it, and yet ultimately, God will do away with all suffering (Rev 21:4-5)…and He will wipe away every tear from their eyes; and there will no longer be any death; there will no longer be any mourning, or crying, or pain; the first things have passed away. And He who sits on the throne said, “Behold, I am making all things new”

We know that Satan is the father of evil. John 8:44 

Sin is the mother of evil. James 1:15 

As for you, you meant evil against me, but God meant it for good, to bring it about that many people should be kept alive, as they are today. Genesis 50:20

God’s control is absolute and, thus, He uses suffering. God uses suffering in the same way that He uses all things; that is, for His glory. John 11:14 

He can use suffering for the glory He gains from using it to bring forth good; that is, good comes out of evil. Romans 8:28

God is not to blame for suffering but He does allow it, and yet ultimately, God will do away with all suffering. Revelations 21:4-5

Answering the Toughest Questions About Suffering and Evil
Bruce Bickel and Stan Jantz

Why Does God Allow Suffering and Evil
 John Ankerberg