So, if we were talking about "this is the Year of the Dog", I think the correct rendering of that in Japanese would be 戌年, with the reading いぬどし. But does that extend outside of the astrological sense?
For example, if TIME magazine declared that "this was the Year of the Selfie", would that seem naturally phrased as シェルフィ年? Would it be better to render it as シェルフィの年? Is there some third, more idiomatic way, to phrase the sentiment "the year when X was prominent"?
The idiomatic way of saying "This was the year of ～" is "今年は～の年だった". Dropping "の" sounds very weird.
And if this Selfie refers to a self-portrait, セルフィー is more natural.
年末を迎えてメディアが指摘しているのが、2013年が "The Year of The Selfie"、すなわち セルフィーの年だったということ。 (http://www.cubeny.com/catch12-3-13.htm)
There is no idiomatic way other than saying 「～の[年]{とし}」.
The 「～[年]{どし}」 pattern basically only applies to the 12 zodiac years. There are however a handful of non-zodiac words you can find in a dictionary, including [厄年]{やくどし}, [当]{あ}たり[年]{どし} ("year of plenty") or [閏年]{うるうどし} ("leap year"), but any of them is associated with or based on some periodic yearly events and unlikely you can coin a new one.
As in naruto's answer, selfie should be transcribed into セルフィー. Wikibooks has a detailed guide to transcribing English to Japanese.
