二十歳 is a (to me) bizarre exception to the usual number+さい rule for discussing age. Is this rooted in 20 being the Japanese age of majority?
Added: To be more specific: why isn't it pronounced にじゅうさい like the rest of the さい words for age?
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The はた there is part of the same series of Japanese readings for numbers as ひとつ、ふたつ、みっつ and so on. Where the ち comes from - that I do not know. It also makes an appearance in some other common words, such as 二十日（はつか）, although in a slightly mangled form.
There are readings for the tens after that as well - for instance 三十（みそ） makes an appearance in words such as 三十日（みそか） and 三十路（みそじ）. The rest of the tens are formed by adding そ to the corresponding "ones" stem: よそ いそ むそ ななそ やそ ここのそ.
Although rarely used these days, the old way of counting was quite flexible. Here's a Chiebukuro question that explains the old way pretty nicely - including how to count hundreds, thousands and tens of thousands!
Some theories from http://gogen-allguide.com/ha/hatachi.html
Please forgive and correct any mistakes I made.
Theory: はた means 20. For example: 二十歳 はたち、二十人 はたとり、二十年 はたとせ。 ち (個)is a counter for the ひと、ふた、み counting system.
Theory (folklore): The 旗乳 (はたち)folktale. During the Warring States period, a young soldier who turned 20 years old wore a banner (旗 はた）of his lords family crest on his back into battle. On that banner he put 20 decorative things (乳 - ち) to match his age. So the theory associates the age of 20 with being old enough to risk your life at war, an adult.
Theory (folklore): If you count your fingers and toes you end up with 20. Deriving from 果て （はて）, you reach the end (はて) at 20.
There are more on that site.