I don't think WotC actually came up with D&D Next; I believe it started as a fan-created hashtag on twitter in response to WotC calling it the next iteration of the game.
WotC made it clear to those that it could reach that they would appreciate it if we would not call it D&D 5E. Primarily because they have no idea yet what the actual name of the next edition will be.
Then a poster started using, as you mentioned above, #DNDNext on twitter (ampersands break hashtags on twitter) and it stuck.