I first heard about these from Professor Tim McGrew:

An undesigned coincidence happens when one part of the Bible is missing a detail that makes us ask a question, and another part of the Bible (a part authored by someone else), or a text external to the Bible, without collusion, supplies that missing detail and answers our question. It is a test for historical authenticity. (Originally published 9/5/11. Added 4/10/13: Video.  Also see Audio Resources by Tim McGrew and Internal Evidence for the Gospels by Tim McGrew on Apologetics315.)

Here are a few examples internal to the Gospels:

(Note: numbers are being skipped on purpose because I am sticking to ‘internal’ coincidences amongst the Gospels and gleaning the examples from here.)

#1:  Matthew 26:67-68  Why ask him to tell them who slapped him?  Luke 22:64  They blindfolded him.

#2:  Mark 6:31  Why are many coming and going?  John 6:4  The Passover pilgrimage.

#3:  Matthew 8:14-16  Why in the evening?  Mark 1:21  Sabbath over at evening (cannot bear burden).

#4:  Luke 9:36  Why did they keep silent?  Mark 9:9  Jesus told them to tell no one (most consistently disobeyed command, lol).

#5:  John 6:5  Why pick Philip?  Luke 9:10  The setting of the miracle is Bethsaida, Philip’s “hometown” (John 1:44).  (And see #17.)

#7:  John 21:15  Why ask “…more than these?”  Matthew 26:33 “Though they all fall away…I will never fall away.”  In John 21:16 Peter is done boasting and just says “Lord, you know that I love you.”  Beautiful!

#8:  Luke 23:1-4  Why Pilate find no guilt in Jesus?  John 18:36  “My kingdom is not of this world.”
Also, John never mentions the charge against him, but Luke fills in the detail.

#10:  Matthew 14:1-2 Why is Herod speaking about this to his servants?  Luke 8:3  Joanna, the wife of Chuza, Herod’s household manager/steward.  Herod knows that if he’s got questions about Jesus, talk to his Christian servants.  Acts 13:1 Manaen had been brought up with Herod the Tetrarch.

#11:  Mark 14:57-58 Mark 15:29  People mock him for saying he would destroy the temple.  He never says that in Mark.  John 2:18-19 Jesus says, “Destroy this temple, and in three days I will raise it up.”

Via email 9/23/11:

#17 “This one just occurred to me: Matthew 11:21 — ‘Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works done in you had been done in Tyre and Sidon, they would have repented long ago in sackcloth and ashes.’ What mighty works in Bethsaida? Matthew gives us no clue; this is the only time he ever mentions Bethsaida. Nor can Mark help us here, nor John. But turn to Luke 9:10 — On their return the apostles told him all that they had done. And he took them and withdrew apart to a town called Bethsaida. … and then comes the feeding of the 5000.” (See #5.)

These are just examples of ‘internal’ undesigned coincidences in the Gospels.

For other types and examples, and more sources for further research, go here.