How can I get Armenian language support in iOS? I find it strange that while the Armenian language (keyboard layouts and characters) is supported on OS X, on iOS it is not supported at all.

A jailbreak on iOS 3 allowed me to install iKeyEx from Cydia, which had an Armenian keyboard in it. This solution no longer works because the app is not supported. Besides, jailbreak probably isn't the answer.
There are 3rd party apps (like iTetr) that can be used for typing in Armenian.

How can I get Armenian language support in iOS?
iOS does have an Armenian font and can therefore display Unicode Armenian in emails, webpages, and text.  For input there are at least two apps that let you construct custom keyboards, and one of them, UniKey, comes with Armenian already:

http://m10lmac.blogspot.com/2011/05/custom-keyboards-for-ios-devices.html

Beyond that or jailbreaking, I think all you can do is ask Apple to add it via their feedback channel.

PS To see Armenian display on your iOS device you can go to

http://www.gov.am/am