Yuan (元) in this sense means "origin" or "primal," intending to evoke a sort of fundamental force behind the Mongol empire, and ideologically undermine dynastic competitors to emperorship of China.

What's interesting about the choice of the term Yuan 元 is that it's the first dynastic title in Chinese history that is ideological rather than a reference to a physical place. All previous unifying dynasties derived their title from the ancient name for the region from which they grew: Qin 秦, Han 漢, Tang 唐, and Song 宋 are all ancient place-names. After the Yuan, the last two dynasties followed Yuan precedent and used ideological concepts as dynastic titles: Ming 明 ("brilliant" or "bright") and Qing 清 ("pure").

Yuan = 元

When the Mongols sinicized themselves, they needed a Chinese name for their dynasty. The character Yuan (元) was chosen from this passage in the I-Ching:

大哉乾元，
萬物資始，
乃統天

(Note the character appears as the fourth character)

My amateur translation of it is:

The great Yang (spring?) energy, 
all things rely on it to grow, 
a universal rule of nature

By naming themselves after the life force that keep nature going, I suppose the Mongol was trying to claim legitimacy to rule over China proper?