What is the primary/basic reason why one has a second chance to bring the Korban Pesach (Passover Sacrifice) on Pesach Sheni (a month after Passover)? Other mitzvos, such as hearing the shofar or other special korbanos (sacrifices) do not have a similar way of making up for the missed opportunity.
The Chinuch (Mitzvah 380) says that celebrating Pesach is so important because it showed the whole world that G-d is in control and powerful, and has the power to renew/create the world ex nihilo. G-d gave us Pesach to celebrate this. Since this lesson is so important, if one missed the opportunity G-d gave him another opportunity to celebrate this.
Actually, it isn't the only one. The korbanot brought on each festival also come with make-up days (7 or 8 days after the onset of the obligation).
Missed tefillah, which were modeled on korbanot, can be made-up for one period after the cessation of the previous period. For example, if one missed Ma'ariv, it can be made up the following morning up until the end of the Shacharit period.
The mitzvah of Milah is primarily the 8th day after birth, however if one misses it it can be made up afterwards. This is referred to as "Tashlumin" by the Rokeach and others.
The mitzvah of Havdalah can be made up for 3 days after the closing of the Shabbat. If one misses the night Seudat Shabbat, it can be made up the next day.
Fasting, if one couldn't do it, can also be made up on any subsequent day, preferably the very next day. (Today it isn't considered obligatory any more, due to the weakness of our generation).
Many other examples exist as well.
The first reason I can think of is purely historical.
The origin of this holiday is an episode recorded in the Torah (Numbers 9:1-14). On the first anniversary of the Exodus, God commanded the Israelites to once again bring a Korban Pesach. There was a group of people who were inelligible to participate at the time due to recent contact with human remains. They went to Moshe and Aharon and asked why they should be prevented from being part of this Korban along with the rest of the Israelites. God responded by commanding them and people who might have this problem in the future to perform a make-up Pesach one month later.
So, you could say that the reason this holiday in particular has a make-up date is because this incident happened when the recurring holiday was just being founded, which didn't happen with other observances.
Of course, you're left with a few issues:
I'm sure that plenty has been written about this, some of which I may try to look up later, but here's how I see it conceptually. The Korban Pesach demonstrates the bringer's membership in the Israelite Nation. One clue to this is in Exodus 12:26-27, where the Torah gives an explicit meaning for the observance: It represents God passing over the Israelite homes, which bore the blood of the Korban on their doorposts, when He smote the Egyptians.
Another clue is in the complaint of the people for whom Pesach Sheini was apparently founded. They specifically say "Why should we be prevented from bringing God's Korban ... among the Children of Israel?" Apparently, the tie to the community represented by this Korban was an important part of their concern.
So, I suggest that both these people's motivation and God's was to make sure that this central rite of affiliation was available to every Israelite.
May the Israelites of today once again demonstrate to God that we can't bear to be prevented from bringing the Korban Pesach, and may He respond by once again giving us an avenue for bringing it.
Perhaps it is related to the fact that eating the korban pesach is one of two positive mitzvot whose transgression gives karet. The other such mitzvah, milah, can also be made up at a later time.