Revenue per search session is total revenue from sessions containing a search divided by the number of those sessions.
It collapses conversion rate and order value into one number, which makes it the cleanest metric for arguing about search in financial terms. Improvements in relevance, zero-result recovery and result layout all surface here.
Tracked per query, it also identifies which searches actually earn — usually a short head of terms worth merchandising by hand, and a long tail worth fixing systematically.