זֵכֶר או זֶכֶר - Zaykher or Zekher

There is no halakhic requirement to repeat the words תִּמְחֶה אֶת-זֵכֶר עֲמָלֵק in Parashat Zakhor.

The origins and history of what has become common practice, to repeat the verse and pronounce the key word as "זֶכֶר" -- with a double segol, are summarized in a 1998 article by Yosef Peretz, based on the work of M. Breuer and Y. Penkower.

In short:

Radak (R. David Kimhi, d. 1235) was the first to claim to have seen both זֵכֶר (zaykher) -- "five dots" -- and זֶכֶר (zekher) -- "six dots" -- in various biblical manuscripts.

R. Issachar Baer of Vilna reports in Maaseh Rav (134), the earliest anthology of the Vilna Gaon's ("GRA") customs, that the GRA pronounced זכר as zekher when he recited the verse on Shabbat Zakhor.  However, the author immediately cites the contradictory testimony of R. Hayyim Volozhiner -- the GRA's close disciple -- that the GRA in fact pronounced the word as zaykher, the same vocalization 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 variations.

However, the oldest and most reliable versions of the Masoretic text of the Bible all have זֵכֶר -- 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 -- in the entire annual cycle of Torah readings, this is the only instance of a repeated variant pronunciation -- 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.

Closely related to this discussion is ArtScroll's usage of zekher in the Ashrei prayer, in every edition of their Siddur (the Chabad Siddur also uses this pronunciation).  This too is based on a purported custom of the GRA, as recorded in Maaseh Rav (28), and here too R. Hayyim Volozhiner refutes the testimony of R. Issachar Baer.  

There is no textual basis for this vocalization, either in the Masoretic Bible or in any variant manuscript, all of which have זֵכֶר in Psalms 145:7.  Like the zaykher/zekher repetition of Parashat Zakhor, the ArtScroll Siddur imposes an obscure practice on mainstream traditional Judaism, irreverently pushing aside centuries-old liturgical tradition.

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Postscripts - אחר כך מצאתי:

See R. Menachem M. Poliakoff, Minhagei Lita - Customs of Lithuanian Jewry (Baltimore: 2009), p. 53:

Despite the suggestion of the Mishnah Berurah to repeat the word zeicher to accommodate the opinion that the reading is zecher, in the Telshe Yeshivah and most other places in Lithuania they did not repeat it.  In fact, according to former talmidim, they did not observe this custom even in Radin, where the Chafetz Chaim (author of the Mishnah Berurah -dsz) had his Yeshivah.

My humble opinion is even if this repetition is permitted, it is certainly bad practice because it publicly introduces an element of doubt into the accuracy of the mesorah (emphasis added).

See, however, R. Hershel Schachter, Nefesh Ha-Rav (Jerusalem: 1994), p. 111 and Yom Kippur Machzor with Commentary Adapted from the Teachings of Rabbi Joseph B. Soloveitchik (New York: 2006), p. xxxv.  Rabbi Soloveitchik is said to have repeated the verse beginning זֵכֶר in Ashrei with the alternate vocalization (note that the positions attributed to Maaseh Rav and R. Hayyim Volozhiner in the Yom Kippur Machzor are incorrectly reversed).  With rare exception, Rabbi Soloveitchik did not recommend his personal liturgical or halakhic practices to his students.  See Schachter, Nefesh Ha-Rav, pp. 3-4.