Most manuscript variants–that is, different words appearing in different ancient copies of parts of the Bible–are theologically insignificant. They change some bit of grammar or other, but don’t affect the meaning. When it comes to Romans 6:12, however, this is not the case.
The majority reading of the manuscripts–the one that appears most commonly and in the oldest manuscripts–reads like this: “Do not, therefore, let sin reign in your mortal body so that you obey its desires.” The ‘its desires’ part refers here to the body, the body’s desires. (It’s impossible to tell in English, but the Greek pronoun αὐτοῦ–meaning ‘its’–refers to the neuter word ‘body,’ not the female word ‘sin.’) To obey the desires of this ‘mortal body’ is to give into sin and allow it to ‘reign’ over us.
But this verse is worded differently in some manuscripts. Some say, “Do not, therefore, let sin reign in your mortal body so that you obey it [the female αὐτῇ, referring to ‘sin’]”: do not let sin reign by obeying it in your body. Others say, “Do not, therefore, let sin reign in your mortal body so that you obey it [sin] in its [the body’s] desires”: to obey the body’s desires is to obey sin–meaning, in effect, the same as the majority reading.
The most interesting variant is one that appears in only a single manuscript, a lectionary from the eleventh century. Here Romans 6:12 reads, “Do not, therefore, let sin reign in your mortal body so that you obey it [sin] in its [the female αὐτῆς, sin’s] desires.” What is fascinating about this reading is that the body is not the source of sinful desires. Sin can reign in our ‘mortal body’ if we let it, if we ‘obey sin in sin’s desires.’ But it has no power over or connection to our very body’s desires, as in every other manuscript.
Of course, this interesting reading is not the correct reading. There are good textual–it only appears once, in the eleventh century–and theological reasons for rejecting it. Ever since our first parents’ sin, sin, like death, has ‘come into’ humanity (Rom. 5:12). It is no longer an external word tempting us from outside, but a force within us, stirring up desires that bear the ‘fruit’ of ‘death’ (6:21). Even after we have ‘died to sin’ in baptism (6:2), it is still the ‘sin living in me’ (7:17) about which Paul cries out to God, “Who will rescue me from the body of this death?” (7:24). Only after the resurrection will it be true that our bodies themselves will be free of sinful desires.