What does Lumosity's Flexibility measure? One of the Lumosity BPI measurements is termed Flexibility.  What is this measuring?
I believe they are attempting to measure Cognitive Flexibility (there is a citation linking to Lumosity at the very end of the article).
From the Wiki article:

Cognitive flexibility has been described as the mental ability to switch between thinking about two different concepts, and to think about multiple concepts simultaneously....
Other terms for cognitive flexibility include mental flexibility, shifting, mental set or cognitive shifting, task switching/shifting, and attention switching/shifting....
Most commonly, cognitive flexibility refers to the mental ability to adjust thinking or attention in response to changing goals and/or environmental stimuli....
Lastly, the ability to simultaneously consider two aspects of an object, idea, or situation at one point in time refers to cognitive flexibility. According to this definition, when sorting cards based on specific rules, children are considered cognitively flexible rules if they can sort cards based on the color of the objects and type of objects on the card simultaneously. Similarly, cognitive flexibility has been defined as having the understanding and awareness of all possible options and alternatives simultaneously within any given situation.

I hope that helps!