In The Hunger Games, Katniss' district (neighborhood) is nicknamed "the Seam." Katniss lives in District Twelve. District Twelve is not the last district, there are actually thirteen districts, but it is the last inhabited district.
District Thirteen was identified as the district which created nuclear weapons. During the Dark Days, District Thirteen was obliterated as a warning to the other districts. District Thirteen tried to fight against the Capitol and the Capitol destroyed the district as a warning to the other districts in order to show them what would happen if they tried to revolt.
Therefore, District Twelve (the district which Katniss lives in) is the last functioning and recognized district. This fact justifies the district's nickname of "the Seam."
A very simplistic definition of the word seam is a line. The Seam, therefore, is the line between all of the districts and the lands not part of the district. All lands beyond "the Seam" are off limits. The Seam, therefore, is the farthest boundary line and is denoted as such by the nickname.
The above answer certainly covers the reason why district 12 is called, "The Seam." However there is an expanded thought here. Foreshadowing is a device that many talented writers use to tell a reader (one who is reading closely) what might be coming down the road. It is not revealed until later that District 13 is in fact still there, in an interesting sort of existence.
So, the answer above is both logical and insightful; this is merely an additional thought to help justify the author’s use of the term, “The Seam.”