This places it at the same "full moon" time of the month as Pesach, Sukkos and Tu B'Shevat. For whatever reasons, that seems to be prime Yom Tov time.
It allows time to become Tahor via Parah Adumah, plus time for traveling. Realistically, that would mean allowing at least two weeks.
It avoids problems (inside Israel) of figuring out the date of Rosh Chodesh, which would be a problem if it were set for an early day in a month).
Waiting until after Shavuos (e.g., 14 Sivan) would definitely not make sense as Shavuos is Atzeres of Pesach.
and of course having it exactly the same day (as opposed to 13 or 15 or 16 Iyar) makes it easy to remember. At least as long as you understand that the date is based on bringing the Korban - and that Erev Pesach is a Yom Tov of sorts as well (just usually quite hectic due to the Pesach preparations and no obvious "no Tachanun" like Pesach Sheni has).