Lindsey Graham is making noise by proposing an amendment that would deny U.S. citizenship to children of illegal immigrants.
Such an amendment might not, in fact be necessary. The Fourteenth Amendment says that "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside." The Supreme Court, if memory serves, has not ruled on whether illegal immigrants are "subject to the jurisdication" of the U.S.
Moreover, as scholars like John Eastman have noted, the Supreme Court was mistaken when it declared that the Fourteenth Amendment awarded citizenship to the children of foreigners.
When pressed about whether Indians living on reservations would be covered by the clause since they were "most clearly subject to our jurisdiction, both civil and military," for example, Senator Lyman Trumbull, a key figure in the drafting and adoption of the Fourteenth Amendment, responded that "subject to the jurisdiction" of the United States meant subject to its "complete" jurisdiction, "[n]ot owing allegiance to anybody else."  And Senator Jacob Howard, who introduced the language of the jurisdiction clause on the floor of the Senate, contended that it should be construed to mean "a full and complete jurisdiction," "the same jurisdiction in extent and quality as applies to every citizen of the United States now" (i.e., under the 1866 Act). That meant that the children of Indians who still "belong[ed] to a tribal relation" and hence owed allegiance to another sovereign (however dependent the sovereign was) would not qualify for citizenship under the clause. Because of this interpretative gloss, provided by the authors of the provision, an amendment offered by Senator James Doolittle of Wisconsin explicitly to exclude "Indians not taxed," as the 1866 Act had done, was rejected as redundant.
Eastman gives a good account of the broader argument. It is well worth reading his full essay. The Supreme Court need only apply the law that the people ratified in order to do what Graham wants to do with an amendment.