There is no real halakhic basis for repeating the words תִּמְחֶה אֶת-זֵכֶר עֲמָלֵק in Parashat Zakhor.
The origins and history of the current practice, to repeat the verse and read זכר with a double segol, are summarized in a 1998 article by Yosef Peretz on a Bar Ilan website, based on the work of M. Breuer and Y. Penkower.
The story is briefly as follows:
The Radak was the first to testify to seeing both zaykher ("five dots") zekher ("six dots") in various biblical manuscripts.
The Maaseh Rav of R. Issachar Baer of Vilna, the earliest anthology of the Vilna Gaon's minhagim, claims that the GRA pronounced the word זכר as zekher when the GRA himself recited the verse on Shabbat Zakhor. However, R. Issachar Baer immediately adds the testimony of R. Hayyim Volozhiner -- the GRA's close disciple -- that the GRA in fact read the word as zaykher, as is found in all standard chumashim; i.e., five dots, not six (R. Hayyim repeats this claim in a haskamah for Maaseh Rav dated 1817). The Mishnah Berurah recommends reading both pronunciations.
However, the oldest and most reliable versions of the Masoretic text of the Bible all read zaykher -- with five dots.
The custom of repeating זֵכֶר עֲמָלֵק only became widespread in the late twentieth century. It seems to me that out of respect for the Masoretic text -- this is the only example of a repeated variant pronunciation in the entire Torah reading -- and to avoid what is most likely a מנהג טעות, there is far more reason not to repeat the verse than there is to repeat it.