Access to information is already a right - this is not controversial.
Literacy is required for that in the modern age so literacy education is also a right ... we all have the right to be able to read and right and to exercise these skills... comes under the concepts of freedom of speech and expression.
However, nobody is out campaigning that access to clay tablets and styli should be a right - they are "tools for writing". This is because clay tablets are not very important to modern written expression ... we'd place more emphasis on access to paper and ink (the 1765 Stamp Act, USA, comes to mind).
In the 21st century, the internet is emerging as a major means of access to all kinds of information - easily as important as paper in the late 1700's. The primacy of the internet in many countries now means that, if not now, then in the near future, access to the internet will become essential to the exercise of existing rights . This is the basis for the claim that internet access should be (formally recognized as) a right in law as well ... meaning that it should be protected in law as a way to protect existing rights into the future.
This is hard to fit on a banner - so the slogan is that "the internet should be a right".
The phenominon of the internet pretty much embodies article 19 of the UN UDOHR which includes the freedom to ... receive and impart information and ideas through any media and regardless of frontiers.
Access to the internet is already an International Human Right.
There are legal implications for this or any country whose powers want to be able to restrict internet access without due process.
Even those who have moral reservations about the rights status of the internet, if they care about due process, and access to information, want to support this idea.