Although there are differing views on what follows, the most convincing explanation I've heard regarding this reference centres around the nature of the culture Timothy was contending with at Ephesus. For many years, Ephesus had been an entrenched and powerful centre of the cult of Ephesian Artemis - her temple was one of the seven ancient wonders of the world, and as is testified to in the book of Acts, the temple 'economy' was so central to the life of the city, that the rise of a competing religion that undermined its support was riot-triggering. Following are some of the beliefs of this cult:
Artemis was like the “great mother” goddess, who was regarded as the universal mother of all life. The Ephesian Artemis was believed to have the power to bring new life into the world and to take life away. It was also believed that Artemis helped women and animals in labour. Ephesian women would call on Artemis during childbirth to speed up the labour and ease the pain, or, in dire circumstances, they would call on her to bring about a quick death to end their suffering. the original human was a woman. - source (emphasis added)
The priestesses in the temple had a highly significant role and most likely had civic leadership responsibilities as well as their spiritual role, which would probably have included gnostic notions of being custodians of the 'mysteries' - ie they would have been the sources of spiritual knowledge. Additionally, one of Ephesus's foundation myths was that it was settled by the Amazons - the legendary war-like matriarchal tribe who would reputedly hack off or cauterize one breast to facilitate their martial prowess. The persistance of elements of matriarchal culture in Ephesus is the most likely reason for the curious nature of the reference and it's application. Paul is encouraging Timothy to disabuse the new converts from any false notions of female superiority that may have carried over from their culture. It was to be emphasized that, women were not to usurp authority over men on the basis of 'being created first' - in fact, the Biblical account affirms exactly the opposite! Likewise, since the mother of all humans was actually deceived, it would be wrong to advance women as being automatically fitted to spiritual leadership and being disseminators of spiritual knowledge on the basis of their gender - as new converts, first and fore-most, they needed to become students of the word of God. (cf. verse 11)
If this is the true context, then it would be evident that an over-generalized interpretation of the preceding verse (12) to the quoted passage to disqualify women from any role in teaching or leadership in the church for all time would be a gross over-reaction that actually feeds into errors from the opposite extreme of patriarchy.
In summary, Paul quotes Genesis in order to counter erroneous notions of female superiority that were present in the Ephesian culture.