The Bible does not identify the "elect lady" directly. The most natural meaning of his salutation is that John sent his letter to a particular person. However, he may also have been referring to the church symbolically.
Although church history does not say to whom John wrote, his letter may have been addressed to Mary the mother of Jesus. Mary's unique role and the special association between herself and John would make it unlikely that he would have referred to any other person in such a way. As Jesus hung dying on the cross, He charged John with His mother's welfare (John 19:26-27). We also know that Mary had several children in addition to Jesus (some are mentioned in Matthew 13:55-56) and a sister (John 19:25) in the church. This is in accord with II John 1, 13.
It would not have been out of place for John—while writing the letter specifically to Mary—to have addressed the church as a very special lady. As shown by the contexts of Ephesians 5:32 and Revelation 12, the church is sometimes referred to as a woman (see also Galatians 4:26). At the time John wrote, false Christianity was on the increase, and God's "children" needed a mother's—the church's—spiritual guidance and protection more than ever. "The children of your elect sister" (II John 13) would then, under this symbolic interpretation, be the members of the church in the area from which John was writing.
While there is a natural curiosity to know the meaning of the "elect lady," we should remember that it is really of no great significance as far as day-to-day living is concerned. The important thing to keep in mind is that this letter is a part of the inspired Scriptures and contains a vital message for Christians today.