Saints definitely aren't Christianized polytheistic deities, though there may seem to be some big conceptual similarities. It's easy to make the jump from many gods ruling many areas of influence to one god with many saints doing the same thing, but doing so obscures a lot of the nuances of the situation.

Saints, first of all, are explicitly human. That's the entire theological justification for having saints: they have lots of cool supernatural powers (working miracles, taming natural forces, etc.) but these are a function of the larger supernatural power of God, not something innate to the individual person. This is also what makes saints so special, above and beyond the average virtuous individual. This is true even of non-human saints, i.e. angels; as early as the Letter to the Colossians (2:18) Christian authorities wanted to make it very clear that saints, even angels, were just a kind of conduit, not a source of power/worthy of worship on their own. More than that, though, saints are often actively opposed to pagan remnants of society. Hagiographies show them putting an end to local cults (e.g. Life of St. Martin 13, Life of St. Nicholas 14-5). When temples were converted into churches, the church might be re-dedicated to a thematically-appropriate saint. This happens a lot for the Virgin Mary: the Parthenon, in Athens, and Santa Maria sopra Minerva, in Rome, were both temples to Athena/Minerva that became churches dedicated to Mary.

Some saints might be a little more ambiguous. Chonai (ancient Colossae, the destination of the letter to the Colossians) was a huge cult center for St. Michael after the advent of Christianity, but before that it had important ties to the cult of Attis. Both St. Michael and Attis are depicted as youthful (handsome, beardless, etc.) and as the attendant of an important female figure (Mary and Cybele, respectively), and there's some thought that Michael's cult might have been especially encouraged at Chonai because it was familiar enough to win over the local pagan populace. The arguments for this make sense to me, but I can't think off the top of my head of similar cases anywhere else. Angels also occupy a weird place in regards to other saints--everyone agrees that angels are basically good, but how can things that never lived human lives be used as human role models?--so I'd hesitate to apply the case of St. Michael too broadly.

I'm most knowledgeable (as is probably obvious) on the theological side of this argument, not the cultural. There may be a broader context for these theological justifications that aligns them more closely with the simple logistics of how conversion to Christianity took place--Christianity was so different from the belief systems that preceded it that I imagine having something like saints and saying "they're kind of like what you knew under paganism, except Christian, and thus better for your immortal soul" must have been very attractive. In the meantime, if you're interested in reading more about this, there are two very recent books that I'm deriving most of this answer from that may be of interest to you:

• Kevin Madigan, Medieval Christianity: A New History (New Haven, CT: Yale University Press, 2015), esp. p. 320-342 ("Devotion: Saints, Relics, and Pilgrimage"). Madigan gives a good general overview of medieval Christianity in general, so the saints are more or less everywhere in this book, but this chapter specifically looks at what role saints played in the grand theological scheme of things.
• Robert Bartlett, Why Can the Dead Do Such Great Things? Saints and Worshippers from the Martyrs to the Reformation (Princeton, NJ: Princeton University Press, 2013). This is a really great outline of what exactly saints do (and, by extension, what they are/are not, e.g. gods); I really recommend it.

I know some of this stuff can get pretty esoteric, so if I can elaborate on anything I wrote here, or if you have any other questions, just ask.

Hey! In addition to /u/arivederlestelle's great answer, you may be interested in a similar question I answered a while back. While the cult of the saints certainly fills a niche once serviced by pre-Christian cults ,that of the supernatural patron, I would not go so far to say it was a direct channeling or a co-opting of Mediterranean polytheism. At least as far as origins and the early centuries are concerned, the practice has more in common with late ancient funerary rituals than the rituals surrounding the cults of heroes and gods in Roman society. The emphasis on death and suffering as a vehicle for obtaining proximity to the divine was also an important break from earlier traditions. In a similar vein, the Christian habit of partitioning relics, parading them around, keeping them on their person, etc. deeply disturbed 'pagan' contemporaries.

At times the cults of saints do seem to take on a role in popular religion once occupied by gods and other supernatural beings. However, I would argue this has more to do with religious and cultural syncretism than a deliberate and coordinated attempt to make Christianity more palatable to the masses. Saints seem to fill a social and cultural niche as supernatural intercessors, a source of 'magic' that individuals might be able manipulate to their advantage. The manipulation of the supernatural was nothing new in the late ancient-early medieval world, but for various reasons, many gradually adopted Christ and his saints as the more viable or potent option. The extent to which the theological debates of the day mattered to everyday converts is often difficult to discern, but conversion was usually not an either/or decision; people typically did not (and don't) discard a lifetime's worth of cultural vocabulary, so it is not surprising that at times saints seem to adopt the vocabulary and tropes of their more 'pagan' predecessors.

The cult of the saints was something new, both in the way it was articulated by Christian theologians and the manner in which it challenged long established taboos regarding death and access to the divine, but that does not mean it can be divorced from the cultural and religious context it was heir to. However, the cult of the saints and those who wrote about it were not a monolithic hive-mind and the degree to which saints resemble and adopt pre-Christian traditions varies from author to author, text to text, saint to saint, and region to region. Some saints come across as very human, and others more directly embody pre-Christian antecedents (see Brigid of Kildare and the Irish goddess of the same name). Saints were not part of a conspiracy to appropriate Roman gods, that misses the nuance of the cults and ascribes to the late ancient church a level of coordination that did not exist. Rather, when saints adopt pre-Christian elements and tropes, it is evidence to the malleability and cultural syncretism that were part and parcel of late ancient and early medieval religion.