Tim’s wife Lydia explains the concept of undesigned coincidences on her blog:
Undesigned coincidences in the Gospels … is an argument that was well-known in the nineteenth century but has, for no really clear reason, simply been forgotten as time has gone on. It is a cumulative case argument that the Gospels reflect, to an important extent, independent knowledge of actual events. Please note that this argument is quite independent of one’s preferred answer to the synoptic question. That is to say, even if, e.g., Mark was the first Gospel and others had access to Mark and show signs of literary dependence on Mark, the argument from undesigned coincidences provides evidence for independent knowledge of real events among the Gospel writers. There are many more of such coincidences beyond those given in the talk.
Basically, this argument finds cases where the same story is in two sources, but where some important detail is left out of one account so that something about the story seems out of place. But the other source has the missing detail that unlocks the mystery. This makes the sources appear to be independent, especially as more of these coincidences pile up. If this happens a lot, it argues for independent sources, which means that the story is multiply attested, which it is it less likely to have been made up.
My favorite example was the Philip example from John 6.
As I was listening to Tim’s examples, I was struck by all the reasons there might be for a real eyewitness not to fill out the explanation for a detail. Think for example how tedious it is to listen to someone who goes back to explain every little detail he mentions in a story.
[...]Similarly, as John is telling the story about the feeding of the five thousand, it would be quite natural for him to say that Jesus asked Philip where they could buy bread if he were really an eyewitness–that is, because he remembered that Jesus did ask Philip. (Tim talks about why it was Philip in the interview.) But John himself might have had to stop and think for a moment if someone had asked him, “Why did Jesus ask Philip rather than any of the other disciples?” Presumably when John told the story, he wasn’t particularly thinking about some special reason for Jesus to select Philip for the question. But if someone were forging the story as fiction, he would have a reason for choosing to use a given disciple as a character at that point in his fictional narrative, and therefore he would be unlikely to choose that character without making the reason clearer to his readers.
All sorts of such things can happen when one is telling a true story, especially a story one has witnessed. One gets caught up in what one actually remembers and drops in incidental references to small facts, which facts are to some extent selected randomly by the memory as one brings the scene back to memory. This is typical of real memoirs but not of elaborate forgeries.
Lydia is also a philosopher, and her Ph.D is from Vanderbilt University. She’s put together a nice list of resources on historical apologetics. She is a homeschooling mom – I LOVE THAT!!!