The period of Ben Ha'shemashot, which extends for a short while (approximately 10-13 minutes) after sunset, is regarded by Halacha as a time of "Safek" – uncertainty, as it is questionable whether this period is to be considered daytime or nighttime. This uncertainty affects many different areas of Halacha, including the date of a youngster's Bar-Misva, the day when he becomes obligated in Misvot. If a child was born during Ben Ha'shemashot, it is uncertain whether he is considered to have been born during the day, or born during the night. As the Halachic day begins in the nighttime, his birthday is uncertain. Thirteen years later, then, we are unsure which day is the day when he becomes a Bar-Misva and is obligated in Misva observance.
Due to this uncertainty, he must begin strictly observing the Misvot on the first of these two days. Less obvious, however, is whether he may serve as a Hazzan in the synagogue on the first day. We would intuitively assume that since he might not be a Bar-Misva until the second day, he should not serve as Hazzan on the first day.
However, Hacham David Yosef, in Oserot Yosef, asserts that the boy may serve as Hazzan for Arbit on the night of the first of these two days. He notes that there is a view in the Gemara that conclusively regards Ben Ha'shemashot as daytime, and, additionally, Rabbenu Tam (France, 1100-1171) maintained that halachic sunset occurs 72 minutes after the moment that we consider sunset. According to both these opinions, this boy becomes a Bar-Misva on the first day, thus giving us room for leniency at least during Arbit, when there is no repetition of the Amida, and thus the Hazzan is not fulfilling any sort of obligation on behalf of the congregation. Hacham David goes even further and posits that there is a basis for leniency even during Shaharit and Minha on the first day.
In conclusion, then, if a boy was born during Ben Ha'shemashot, and thus the precise date of his birthday is uncertain, he becomes obligated in Misvot on the first day, and may serve as Hazzan for Arbit on the night of the first day. If he wishes to serve as Hazzan also at Shaharit and Minha on the first day, there is a basis for allowing him to do so.