For similar reasons to those posted above, I would also say the answer is yes. Emily's status as the last in a family line is important to the meaning of the story. As the world changes around Emily's house, she clings to the vestiges of the past, refusing to change along with the world around her. She is the end of something, representing a whole culture that is passing away.
Yes it is significant that Miss Emily is the last of the Griersons. Faulkner makes mention of "the august names"--those respected and venerable families that used to populate the region. Emily is the last, and she is no longer respected or venerable. In addition, look at how her life has slowly arrived at it's lonely and unsavory conclusion. This is not the way we would expect one of those "august" families to end.
Miss Emily is the last of her kind, certainly. She is the last living Grierson, of course, and she does represent the dying vestiges of the Old South, as the posts above point out. When Emily appears to be "living in sin," she is visited by relatives who also hold the values of the Old South. Though she may be the last Grierson, she may not be the last representative of the Old South.
Faulkner's "A Rose for Emily" is, in his own words, "an allegorical tale." Therefore, it is most assuredly significant that Emily is the last of the old Southern gentry name, Grierson. Emily is not so much an individual as an allegorical character representative of the patriarchal society of the antebellum South. This is why the old Civil War veterans feel that they have known her and perhaps have danced with her; in Emily they recognize the last vestiges of a era gone by, an era of their youth that they romantically glorify.
The fact that Emily is the last of her family does not really impact the meaning of what you might call the "personal level" of the story. This is the part that has to do with Emily's own motivations. I think that Emily is concerned mostly with her inability to find someone to love and to share her life with her, not with perpetuating her family line.
However, I agree with the first post that there is another level to the story. On the more societal level, it is imporant that Emily is the last Grierson since she does represent the death of the Old South.
Miss Emily's passing as the last of the Grierson family fits into several of the themes--particularly those of death and the decline of the Old South--that William Faulkner discusses in "A Rose for Emily." There are a number of references to death throughout the story, and Emily's desperate desire to marry might, in some manner, be in order to continue the family name (though it seems to be primarily for her own happiness). Her death as a single woman and the last of the Griersons certainly fits symbolically with the decline of Jefferson in the post-bellum decades following the Civil War. Like the few remaining war veterans of Jefferson who dwindle with every year, Emily's death--and the end of the Grierson line--symbolizes the end of the Old South and the gradual rise of the New South.