There are several related terms/concepts

- Seeing special meaning in unrelated things (like: face = meaning, unrelated = rocks in a random pile) is ['aphophenia'](https://en.m.wikipedia.org/wiki/Apophenia).

  - When it it is recognising faces then it is called ['pareidolia'](https://en.m.wikipedia.org/wiki/Pareidolia). 

  - A special case related to statistics/randomness is the ['clustering illusion'](https://en.m.wikipedia.org/wiki/Clustering_illusion). Where clusters or patterns in random distributions are considered as unlikely.

- Creating arguments that lead to a predetermined conclusion could be ['begging the question'](https://en.m.wikipedia.org/wiki/Begging_the_question).

- The behaviour to look for causal explanations in coincidentally occuring events is called ['synchronicity'](https://en.m.wikipedia.org/wiki/Synchronicity).

- The ['look elsewhere effect'](https://en.m.wikipedia.org/wiki/Look-elsewhere_effect) is the effect that the probability of a significant result is increased when we look multiple times for such significant result.

In the comments you state that the look elsewhere effect is what you were looking for. But that is solely the fallacy of a false expression of the probability of an event/effect occuring (by considering a single test and ignoring multiple times looking in other tests). 

The assignment of a special causal meaning to such improbable/random event (which actually was probable, or even when it was improbable even when accounting for the look elsewhere effect, then it might still be a random event) is more related to aphophenia and synchronicity.