Djinni are almost certainly a pre-Islamic set of entities, but their "survival" of conversion wasn't really a trial. Djinni were and still are an accepted part of the created order of the universe. They didn't really need to be converted any more than sand needed to be rethought. Within Islamic thought, the Djinn are basically just like humans. They go around, do their own thing, and exist just like we do. The Thousand Nights and a Night are a prime example of this. Most of the stories themselves are almost certainly pre-Islamic in origin, but they were circulated even under a Muslim paradigm. There are djinni who served Satan, there are djinni who help people, and there are djinni who just run around and mess things up.

Robert Hoyland in his book refers to north Arabian accounts of "ginnaye" as tutelary deities whom are believed to provide protection for local communities in pre-Islamic period. Related to these are the "jinn" featuring in west and central Arabian accounts. In the Quran, the jinn are said to be worshipped by people; however, confronting the pre-Islamic belief, they are said to be created beings like humankind. Robert G. Hoyland; Arabia and the Arabs: From the Bronze Age to the Coming of Islam, pp.144-5, 2001, Routledge.

I've looked this up before but couldn't find an answer: is there any connection between djinn and roman genii? They just seem to have pretty similar consonants.