Atheism had no part in designating Human Rights, which is why it's hard to argue them as one.

Let me explain the path a lot of modern human rights come from. The big ones you think about, at least.

The United States were founded on an appeal to divinity pitched deliberately in general, non-sectarian terms so it stayed interpretable across a religiously plural people and kept the state out of doctrinal disputes. HOWEVER, the general frame itself was substantially the product of Christian moral thought, and for most people of the time it was naturally understood in specifically Christian terms.

So, the argument that "The US is a Christian Nation" is only true in two ways. It's true today via demographics, and it's slightly true in its founding ONLY by also looking past the direct words of the constitution and reading it in the period's context. There are no direct Christian references in any early founding documents of the US. 

Here's the big thing though. it was not to be an ATHIESTIC nation. It is a nation inclusive to all religions, but it is very clearly stated that the rights are derived via divine morality, given to man by its creator. See the quotes "the Laws of Nature and Nature's God," "endowed by their Creator," the Supreme Judge of the world." Every single one of these suggests what I argued. It's for this reason that I believe lawmakers should still look at morality as something sacred instead of something non-existent or super "flexible". Sure, person-to-person it may vary, but the nation itself needs consensus as it was consensus on morality that led to its very founding. 

I don't want to pull credit from Christianity in its founding in any way. Christianity was arguably the most important factor in the appeals of the declaration and constitution, specifically the Christian idea of Natural Law. Christians literally created the moral law that regulates ALOT of the world these days, with atheists often not even knowing this! There is a lot of common ground!

In the 1948 Universal Declaration of Human Rights, the opening move "the inherent dignity and equal and inalienable rights of all members of the human family" is literally imago Dei rewritten in a diplomatic way. Nearly every right below flows through it.

Inherent, equal human dignity. - Genesis 1:27

Slavery is an evil. - While defended by Christians for centuries, it was ALSO Christians that pushed for its end. The ultimate ruling was that the slavery defense was a misinterpretation, allowing less reputable scripture to override the aforementioned scripture, Genesis 1:27. Basically, the argument was comparable to someone trying to find some small text or caveat in a terms agreement to defend an action that is literally printed "DON'T DO THIS ITS EVIL" in big text on the front page.

The modern idea of rights attaching to a person, and not a passport - The concept of loving your enemies and pushing obligation past kin, tribe, citizen, and stranger in the western world can be credited to Christianity. While the concept of just "being super nice" isn't created by it, the ideal was absolutely transmitted to the world as we know it via Christianity. 

The biggest hit comes from Nietzche himself. He has written how he saw with total clarity that compassion for the weak, equality of souls, the dignity of the lowly, and the guilt of the powerful were Christianity's fingerprints on western morality. He also states that the reason a secular humanist will call these rights obvious is only because they were raised inside the civilization Christianity built. A modern atheist defending human rights is often arguing in the same exact way a Christian would, while simply crossing out the scripture with whiteout. They stand on moral ground built by the Church.

There are absolutely new concepts that are considered rights by one and not rights by another. These new rights are so brand new and shiny that it's going to be hard to argue them as rights, especially when there's next to no basis beyond "it would be nice for them." When many human rights arguments in the past thousand years are on the basis of religious text, and thus easily debatable between those who share the core belief (see the slavery argument from before), it's almost impossible to argue something as a right when the historical definition of what is a "right" is something directly given by divinity, and not some human-invented concept that needs to be followed because you said so. 

To a religious individual (the majority of the world, mind you), a right's defense being a line of scripture or religious argument tying it to the divine is a much stronger base than "well i think it's a right sooooo... you're wrong." It pulls it out of conceptual thought and into something directly citable. 

It's really doesn't matter what you believe personally, what matters is the fact that you're arguing to a world that DOES believe, and you need to understand that before trying to butt heads with all of them. If a right's basis is simply manmade rules, then the argument against that right has the exact same substance behind it, meaning there is no value in either.