The Wii will record playtime overall for all players and what games were played, but to my knowledge this is not on a Mii-to-Mii basis. Not unless a specific game is programmed to track individual Mii statistics, such as Wii Sports Resort (a fantastic game to get youngins up and about, by the way).
I don't believe such data can be extracted from the Wii without performing some warranty-voiding activities, if you catch my meanin'. Even if that weren't the case, you'd also need some very specific programs in order to read the data. Programs which are, to my knowledge, sadly not available to the public.