What does "nico nico nii" mean?
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.