Answer

The “Eliya rabba” quotes a custom to recite before the end of shmone esreh before one says the “yhyu lratzon”a verse from Tanach which starts with the first letter of ones name and ends with the last letter of his name .

The “kitzur hashloh” explains the reason for this custom so that one will not forget his name at his day of judgment when he reaches the world to come.

Another point found in a extension of Rashis  commentary to the prophet Micha is that one who says a averse connected to his name the torah protects him from gehinom.

Explanation

According to Kabbalah and chassidus the letters of ones name are channels which convey his chayus and life to him through the letters of ones name his spiritual life / neshama connects with his body  when one g-d forbid sins he blemishes his connection between himself and his soul and thereby forgets his name (his connection to his soul) by saying over his verse he will overcome the forces of evil and thereby implement that his life is from the side of holiness so too he will  remember his  g-dly soul and its name when he comes to the world to come.

The Rebbe Rashab related to his son the Frideker Rebbe that when one says the verses with his name he cleans out his soul from negative forces which can effect it being that the soul is enclothed in a physical body,

Nick name

The name which is to be used is the name one uses to have a Aliyah to the torah, For example if one has the name avrohom and everyone calls him Avi nonetheless he should use a passuk which begins with a aleph ends with a mem and not with a yud.

One who has two or more names recites a verse for each name.

Interesting to point out If one’s name is actually found in the verse then that is sufficient and then one doesn’t need say a verse which ends with his letter.

Some chassidim also have the custom to recite the passuk of they’re Rebbe’s name in order that when they come to heaven they will be connected and associated to their Rebbe.

Sources

מקור המנהג הוא ברש”י לספר מיכה, על הפסוק “ותושיה יראה שמך” (ו,ט) הכותב כך – “שכל מי שאומר בכל יום מקרא המתחיל ומסיים כמו שמתחיל ומסיים שמו, התורה מצילתו מגיהינום”. זוהר-חדש רות פד,ב. ספר הבהיר אות פ ובאור בהיר שם, וש”נ. א”ר סו”ס קכב. שערי הלכה ומנהג או”ח ח”א סי’ עג. אג”ק כרך ה עמ’ י. סידור צלותא דאברהם ח”א עמ’ שכז, וש”נ.