well it may seem that the movie's definitely gonna have a 'pit' in the literal sense (jodhpur - green screen as entrance to the pit that people are rappelling down in the trailer, which likely seems entrance to the stair-case-filled underground place that seems to become bruce waynes prison). this prison might be informally referred to as 'the pit', with no connection to the lazarus pit in the comics other than the teasing name.

the most likely scenario to connect the pit with resurrection as has been mentioned before is a metaphorical one. if it has always been a prison, ra's back in the day may have found an ignorant bane in his youth, and/or on the verge of death and come to give him a new life ra ra ra (pun intended), (just like he did to bruce wayne when he was in prison) (so in a sense it could be said that the lazarus 'effect' is an ability of ra's al ghul, rather than a prop)... so the 'lazarus effect' is another word for very effective motivational speaking??

edit: just looked up lazarus effect. definitely a real thing(although obviously not in the way i suggest. but now we know a way the word 'lazarus' can come up in a conversation, and if not, there are synonyms that mean the same thing ('rises', etc).. therefore the words 'lazarus' and 'pit' and what they mean can be referenced in the movie in a realisitc way

one thing im curious about is if they mention india as a country, and if they mention that country being the place that 'the pit' is in. it wouldnt be a far cry because there has been mention of places other than gotham city in nolans batman (south korea, china). But, if it is india, how do they transport all the prisoners (bruce wayne/gothams rich) there, provided that the rumours of imprisoning such people are true?