It is an ancient and established Jewish custom dating back many centuries for a person to recite a pasuk from Tanach which begins and ends with the first and last letter of his name, at the end of the Shemoneh Esrei, prior to the second Yihyu Leratzon. This custom is recorded in many Poskim and Chabad literature. The reason for this custom is because when a person passes away, he may forget his name, and the saying of this verse helps one remember his name, and helps to save him from Gehennom. Likewise, the Rebbe Rayatz states that through saying the verse of one’s name, helps shine the ray of one’s soul into his body, and cleanses it and purifies it from all dirt and filth, and allows one at night during sleep to enter the upper chambers and witness what occurs there.

Sources: See Toras Shalom Rashab p. 116; Igros Kodesh Rayatz 6:273; Siach Tefila 6:5; Piskeiy Teshuvos 122:4; See regarding the general Minhag of reciting the pasuk of one’s name: Rashi on Micha 6:9 on verse “Kol Hashem Lair” [however, see Piskeiy Teshuvos ibid footnote 24 who explains that this is not really from Rashi but from the Sefer Agudas Shmuel]; Elya Raba 122:3; Kitzur Shelah Os Kuf; Siddur Nehora Hashalom; Siddur Rav Naftali Hertz; Yifei Laleiv  Kuntrus Yosher Leivov 122:3; Daas Toprah 122; Kitzur SHU”A 18:15; Likkutei Maharich; Chofetz Chaim Shemiras Halashon 2:7; Kaf Hachaim 122:11; Otzer Hachaim 76; Sefer Haminhagim p.12; See regarding the reason for this custom: Rashi ibid; Siddur Nehora Hashalom; Kaf Hachaim 122:11; Zohar Chadash Rus 84; Parshas Acharei; Chochmas Hanefesh o Rokeiach 7b; Reishis Chochma Saar Hayirah 12:2; Toras Shalom Rashab p. 116; Igros Kodesh Rayatz 6:273; Sidur Im Dach Hearos Utziyunim p. 762