For the noun meaning an indefinitely long period of time, eon is the preferred spelling in American and Canadian English, and it is also preferred by scientists, especially geologists. Outside North America, aeon is favored for uses unrelated to science, but eon appears about a third of the time.
Eon does not denote any specific amount of time (except in geology, where an eon is a period of half a billion years or more). It’s usually used to describe huge stretches of time outstripping even humanity’s tenure on Earth. While the similarly vague era is useful for breaking up periods of human history, eon usually applies to even greater blocks of time. In reference to human events and creations, eon is too big.
These North American publications use eon:
The latest, greatest BlackBerry smartphone, the Torch, is nearly a year old — an eon in the fast-paced mobile device market. [CNN]
The cause of all this disruption is plate tectonics, the slow movement over the eons of separate chunks of the planet’s crust. [Toronto Star]
This series started—eons ago, it seems—with discussion of whether James would end up the best player ever. [USA Today]
And although eon is sometimes used in British and Australian publications, aeon is preferred—for example:
Aeons ago they imprisoned a specially nasty monster called Parallax, who grows stronger on fear. [Evening Standard]
Over the aeons there have been any number of ingenious tricks played by Australian officialdom to stem the tide of human traffic. [The Age]
[Y]ou can witness the spectacle of the Grand Canyon, where aeons of geological time conspire to make winning even the biggest jackpot seem insignificant. [Financial Times]