A group or group action is amenable if it has an invariant mean. Apparently, the term "amenable" is a play of words on "mean" (in British English amenable is pronounced "a-mean-able", hinting at "able to support a mean"), see this MSE posting. Measurable (from the original name in German) would have been a more appropriate name.
With reference to the Brown-Ozawa book cited in the OP, the invariant mean for an amenable action is constructed in definition 4.3.5. See also definition 1.2 of Anantharaman-Delaroche for this construction.