My understanding is that Stoicism was the most influential philosophy in Rome until the late Roman empire, at which point neo-Platonism became more prominent. This seems really weird to me because I don't see what would make neo-Platonism attractive to a typical layman. Did neo-Platonists modify their worldview to make it more suitable for popular consumption?
First off, it is good to note that "Neo-Platonism" is a term applied by more modern scholars to the philosophical "school" of folks like Plotinus and St. Augustine. Plotinus et al, however, would not have called themselves neo-platonists and indeed would probably not even have recognized the term. Indeed, the relevant thinkers could plausibly argue that the label is misleading, since while they do hold the writings of Plato in especially high regard, they are not really engaged in any systematic attempt to revitalize Plato's work. Rather, they are attempting to forge a synthesis between Greek literature, stoicism, platonism, myth, and religion.
As /u/kublahkoala rightly notes, the religious aspect took on especial importance due to the relative harmony that could be established between certain neo-platonic ideas and Christian dogma. But, we mustn't forget that Plotinus, traditionally taken to be the virtual founder of the school, was not a Christian. While his Enneads had a profound effect on later Christian (and indeed Jewish and Islamic) thought, pagans as well as monotheists could be rightly characterized as neo-platonists.
That said, the pairing with Christianity does seem a natural one. One of the central upshots of neo-platonism is the underlying unity of some highest principle. Plotinus, at least in most English translations, refers to this principle as "the one". For a Christian, it is easy to identify this underlying, unified principle with God. Further, this underlying principle in some sense explains (or, to use more contemporary language, grounds) everything else. But, if this is so, then the neo-platonists assumed that this underlying principle must therefore be the most "real", in some sense. Again, there is a natural harmony with Christian dogma; the pure unity of the transcendent God (and, by extension, of the spirit) is a higher reality than the physical realm (including, importantly, the body).
But we should also note that while this seems like a natural pairing, it is by no means essential. The Stoics, for instance, could rightly claim to be at least as compatible, since they too shared a belief in the primacy of the "ruling center" above the body and affirmed the existence of an underlying principle that grounds all else. The writings of Seneca, for instance, were held in high regard by Christians.
Now, to get back to the question of the "popular appeal" I'm unsure exactly what you mean. Abstract philosophical disciplines were typically the province of the well off, and much less popular with the masses than religion. Nevertheless, given the profound influence of St. Augustine, himself a neo-platonist or near enough, it is perhaps understandable that a certain amount of neo-platonist ideas would filter down to the masses via religion.
As for whether the neo-platonists modified their views to make it more suitable for popular consumption, the answer is both yes and no. No, because I think we should give them a bit more credit as earnest philosophers, seekers and lovers of wisdom for its own sake. When St. Augustine or Plotinus defends a position, I think we can safely assume they do so because they honestly believe it to be true, the opinion of the teeming masses notwithstanding. But, I think we can also give something like a tentative yes, given that neo-platonism did, through the work of folks like St. Augustine, get put into something like a firm harmony with Christianity. At this point in history, the achievement of such a harmony was crucial to the lasting success of a school. We aren't that far away from Justinian closing down the pagan philosophical schools, after all.
Some Sources
Andrew Smith's Philosophy in Late Antiquity is a great introduction to philosophy during the relevant period.
Richard Wallis's Neoplatonism is a good intro to neo-platonism in particular.
As for primary sources, the obvious recommendation is Plotinus's Enneads. For St. Augustine, City of God.
And, as always in philosophy, the SEP is a great resource.
A big reason for the revitalization of NeoPlatonism was the rise of Christianity during the declining years of the Roman Empire. Stoicism emphasizes the immanence of daily reality, but for a NeoPlatonist there is a transcendent realm more real and true than our own, allowing it to mesh nicely with religion. A NeoPlatonist would believe that his body was just an imperfect shadow of his true self. A Christian Neoplatonist would probably believe that he had the free will to accept Gods grace ( without which he'd be damned) but God already knew how he would choose. Also, that evil was only the absence of Gods grace, and had no being in itself.