It seems that circumcision was a practice taken over from Egypt by the Israelites. In Egypt, grown men would circumcise themselves or get circumcised, and boys would do this on the border of manhood. This is common in many Ancient Near Eastern cultures. At least one of the goals of circumcision was to protect a person in the netherworld (Lods, A. 1943. "La 'mort des incirconcis'," Comptes rendues des séances de l'Académie des inscriptions et belles-lettres, pp. 271–283).
Although the scriptural sources suggest that 8-day circumcision had been practised for a long time (since Isaac) by the Israelites, that does not appear to have been the case. All these scriptural references are from the P-source (one of the sources of the Pentateuch), which was written in the context of the Babylonian exile in the 6th century BCE.
It seems then that we must explain both a late circumcision (your actual question) and an early circumcision (compared to related cultures). Propp (1987: "The Origins of Infant Circumcision in Israel," Hebrew Annual Review, pp. 355–370) suggests several reasons for an early circumcision on pp. 362–363:
[T]hrough experience the Israelites probably discovered the operation is less painful for babies. Infant circumcision also relieved the individual of the painful choice of undergoing the trauma, for in his adulthood he might have been reluctant. [...]
[C]circumcision may have been considered salutary. Either God pro-
tects the Israelite child, or demons avoid it [...]
Babies were circumcised as soon as it was safe because they were liable to die, for, as Lods (1943) shows, circumcision was believed to improve one's fate in the world of the dead.
And on p. 366 for a late circumcision:
[T]he blood clotting mechanism does not function fully until the age of six months and is particularly deficient in the first two to four days after birth. A few bad experiences would have shown the advisability of waiting a while. But why was the eighth day, i.e., a week plus a day, singled out as the proper time for circumcision? The number eight elsewhere signifies the end of a period of taboo. [...] It seems that the infant is taboo until its eighth day, when circumcision introduces it to the world.
(The part left out contains relevant references, but is omitted for brevity.)
I would like to add the following point to this: during the Babylonian exile, the Hebrews lived in a multicultural environment and felt it was important to establish an identity. It is from this that the strict rules from Leviticus and other P-passages originate. In such an environment, it can be important to "mark" a child as belonging to your group so that he remembers it himself, and feels physically connected to his own group.