Why did God allow Evil into the World?

 No-one Knows  (and that's OK)

“For my thoughts are not your thoughts,
    neither are your ways my ways,”
declares the Lord. 

 “As the heavens are higher than the earth,
    so are my ways higher than your ways
    and my thoughts than your thoughts."

Isaiah 55

True worshippers are never the happier than when they acknowledge the incomprehensible grandeur of our ineffable God in worship: 

Eternal Power, whose high abode
Becomes the grandeur of a God,
Infinite lengths beyond the bounds
Where stars revolve their little rounds:

 

Thee while the first archangel sings,
He hides his face behind his wings;
And ranks of shining ones around
Fall worshipping and spread the ground.

 

Lord, what shall earth and ashes do?
We would adore our Maker too!
From sin and dust to thee we cry,
The great, the holy, and the high!

 

(Isaac Watts)

To worship a God we can rationally understand is to worship our equal and thus to worship an idol. 

Even after the incarnation has revealed so much more of the character and beauty of God there remains much room for wonder:

Oh, the depth of the riches of the wisdom and knowledge of God!
    How unsearchable his judgments,
    and his paths beyond tracing out!
³⁴ “Who has known the mind of the Lord?
    Or who has been his counselor?”
³⁵ “Who has ever given to God,
    that God should repay them?”
³⁶ For from him and through him and for him are all things.
    To him be the glory forever! Amen.

Romans 11

So no believer should be troubled at the ineffable, indescribable, inscrutable ways or character of God. 

The hymn by Watts quoted above ends thus:

Earth from afar has heard Thy fame,
And worms have learned to lisp Thy name;
But, O! the glories of Thy mind
Leave all our soaring thoughts behind.

God is in Heaven, and men below;
Be short our tunes, our words be few;
A sacred reverence checks our songs,
And praise sits silent on our tongues.

A Feeble Potential Answer?

After saying all of the above, and acknowledging that no-one on earth knows why God permitted evil into the world, here is one line of reasoning, which I believe flows from Scripture, if not always chapter and verse, then at least in line with general doctrine.

We know that God is utterly sovereign, that he "works out everything in conformity with the purpose of his will" (Ephesians 1:11). So we know that no independent power rose to rival God, "you can do all things; no plan of yours can be thwarted." (Job 42:2) 

So God more than knew of the fall of the evil one and had good purposes in mind through Satan's fall.  

We also know that the universe glorifies God, "How majestic is your name in all the erth" (Psalm 8), "the heavens declare the glory of God" (Psalm 19).

With this in mind, suppose that the ultimate purpose of the universe is to glorify God - to make him known. This is no selfish or proud end, which would be the case if we wanted ourselves to be known. All our 'glory' is reflected glory; as the moon has no light of its own, so any glory we might possess is borrowed rays from the Sun who should, alone, be praised.

When God sets a purpose to glorify himself, that is in every way a good, noble and proper thing, without selfishness or shame, because he is, in fact glorious.

Suppose the ultimate purpose of the universe is to glorify God - to make known all of his qualities, his attributes, his character. That's what glorify means, to make famous, to reveal, to make known.

Well, in a perfect world, at least some of God's glorious attributes would be unknown. In a perfect world, God's goodness, his power, his greatness could all be known. But not his mercy and grace, nor his justice.

So God allows one of his great angels to fall. God allows that angel to tempt human beings. He allows those people to sin. Now - and only now - is the stage set to reveal the incredible mercy and grace of God. Grace is lavish kindness towards someone who does not deserve it. Before the fall of mankind there existed no "someone who did not deserve it," so the grace of God remained unknown to the universe of angels and men.

So too God's justice. How can justice be revealed if there is no-one deserving of judgement?

Could this be one reason God allowed first the Serpent and then Adam and Eve to fall and to sin? So that he could show grace to those who repent and justice to those who refuse?

And in this way to make known to the universe his justice and grace?

To glorify himself?

 AI Drawing above: "Dalle, draw a painting in the style of digital art  evil coming into the world, where the world is a round globe"