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.
Sources:
- Tractate Menachot 65a and Rashi ibid.
- Tractate Shabbat 119b
- Tractate Berachot 60b, Rashi ibid.
