জ্যায়াই is a contraction of 'yama nashi, ochi nashi, imi nashi'. This translates to 'no (plot) climax, no denouement, no meaning'. It was a derogatory term meant to imply the story had no purpose in existing. However, in জাপান 'yaoi' has fallen out of common use and these days the category is generally called BL ('Boys Love').
Yuri is named after the female character Yuri from Dirty Pair. Dirty Pair was an 80s series of light novels, জাপানি কমিকস মাঙ্গা and জীবন্ত starring Kei and Yuri, two girls who worked as partners. The explosion of Kei/Yuri fanworks had such a significant impact that Yuri's name has been used for girl/girl works since then.