The seven day week is a Jewish creation (even Wiki seems to agree), and we Jews trace this number (which does not work well with either the moon or the sun) to the Torah itself, and the description of creation over a period of seven days. The number is thus quite meaningful to Jews – seven is the number of G-d’s creative acts, the number that culminates in the day we make holy, Shabbos.
Noah is commanded to bring seven pairs of the spiritually ready (King James translates as “clean”) animals into the ark. Why? I think it is because these animals, like Shabbos, are capable of spiritual growth: people can use them as kosher food or sacrifices, spiritually elevating both the animals, people, and the world around us.
So why is Noah told to only bring two of each of the spiritually unfit animals into the ark. I think the number in this case refers to the second day of creation – the only day that G-d does not call “good.” It is not a day of elevation (one form of holiness), but a day of separation and division. The second day of creation was, essentially a stutter-step in the creative process. So the animals that are brought on, in the words of the song, “by twosies, twosies” are the animals that, like the second day itself, do not contribute to the spiritual growth and completion of the world.
For those who are still following, there is an interesting footnote as well. The Torah’s actual language regarding the pair of unfit animals is the word “two”, but the words for the seven pairs are “seven seven”. I think the “seven seven” refers to the notion that there are seven earthly levels, mirrored by seven levels of heaven (I described this in the past comparing the menorah to the corn in Pharaoh’s dream). The animals that are capable of spiritual growth have a spiritual mirror as well, hence the “seven, seven.” The unready animals are merely physical, isolated from spiritual potential in the same way that on the second day, G-d divided and cut off heaven from earth.