This responsum from the "Ask the Rabbi" column of the Ohr Somayach website provides several explanations for the popular custom of reciting each stanza of Shalom Aleichem three times, as opposed to once. Due to its use of Hebrew terminology and halachic concepts, this post may be best understood by those with some background in Jewish learning. Ohr Somayach is a Jerusalem-based, Orthodox learning institution that provides learning and educational opportunities for all types of Jews worldwide.
Question:
Three years ago my mother-in-law asked why we say "Shalom Aleichem" (Peace unto You) on Friday night three times. I gave the reason for being a "chazaka" (emphasis) but that itself wasn't satisfying. Since then I have asked lots of people and nobody ever gave me a different answer.
Answer:
The Likutei Maharich asks your question. He explains as you did, that repeating something three times adds emphasis. We do the same in other parts of our liturgy, such as in kiddush levana and the bedtime shema.
But I've heard another answer: The Talmud states that two angels accompany a person as he walks home from synagogue Friday night. These angels are in addition to the usual angel who accompanies him at all times. Hence, we have three angels in all. Thus we recite "Shalom Aleichem" one time for each angel.
The problem with this answer is that people are usually accompanied by two angels at all times, making four in all. This can be answered according to the Zohar, which states that not two angels but rather two groups of angels accompany a man on Friday night. These, in addition to the usual group of two angels which accompany the person at all times, give us three groups in all. We recite one "Shalom Aleichem"for each group. This also explains the use of the plural "aleichem" – "unto You (plural)" – as well.