Noun for something that is randomly chosen?

Something chosen at random can be called arbitrary.

arbitrary (adj.) - Based on random choice or personal whim, rather than any reason or system

• happenstance (n) A chance happening or event.

http://dictionary.reference.com/browse/happenstance?s=t

• Fortuitous (adj) Happening or produced by chance; accidental

http://dictionary.reference.com/browse/fortuitous?s=t

I can think of

indiscriminately - In a random manner; unsystematically:

or maybe even a less profound alternative like unselective or aimless, depending on the context

I think indiscriminately might be a better choice than arbitrary, because I believe the latter has a connotation of personal choice based on a mood. I am, however, not too sure about this and might be corrected accordingly.

Randomness is the noun form of random, but it doesn't work for usage as "such and such was a randomness".

You can use it in the phrase act of randomness, which may be what you're looking for. Unfortunately I think "of randomness" is essentially just an adjective, and therefore it might not be any better than the many synonyms of random that have been given as answers.