What does "nico nico nii" mean? In Love Live, a character named Nico Yazawa constantly says her catchphrase:


  Nico nico nii!


It seems like if her catchphrase is derived from her own name, at the first glance.  Sometimes, it is extended to a longer line, as she would say, for example:


  Nico-nico-nii~  Anata-no haato-ni Nico-nico-nii~ Egao todokeru Yazawa Nico-nico~ Nico-nii te-oboete, Love - Nico ♡


What does "nico nico nii" mean, exactly?
This is a pun using her name (Nico) and the onomatopoeia for "smiling" or ニコニコ(niko niko)
This is why part of the longer line you have included has Egao todokeru/"delivering a smile".

Edit:
After contemplating a bit longer about this, it dawned upon me that another aspect that ties into the smile part of the phrase is the nii ending. In English when taking a picture, photographers usually use a phrase like "Say cheese!", or in my native tongue, "Say whiskey!". They emphasize the ii sound of ee and y to shape the mouth into a smile.

In Japanese, はい、チーズ(Hai, cheese) is standard as well, but to a lesser extent １+１は？ is also used, the answer being 2, or に(Ni). So the nii part also evokes a smile when saying it out loud.