Jeremy Bentham dismissed the 18th century equivalent of our notion of human rights as 'nonsense upon stilts'. That judgement may be a little harsh, but the notion of rights certainly is in trouble.
For a start there is the problem of 'rights inflation' whereby more and more supposed rights are added to the list, undermining the credibility of the concept.
Then there is the tendency to ascribe rights to people who can't understand the notion (e.g. infants, or the severely mentally disabled), and even to animals. It is at least questionable to speak of a being having a right who cannot understand that he or she has it and to whom it would be absurd to ascribe obligations. The case of infants and children deserves special treatment - they at least will in time assume understanding and obligations.
There is always the option of reverting to traditional ways of speaking, of course, and, for example, simply proscribing cruelty. We can be perfectly humane without rights talk!
Another problem with the rights notion is that it arose in a world with different belief-structures. Rights were seen to derive either from God (God-given rights) or from a metaphysical notion of Natural Law, which is not the same as the 'laws' of science but a transcendent moral system. For those who believe in God and/or something like a transcendent and morally-based Natural Law, the notion of universal human rights makes some sense - but there still remains the problem of knowing exactly what these rights are.
For those who don't believe in such a spiritual realm, human rights are even more problematic. What is their source?
One could say they arose historically and are a product of civilization, but the very fact that the notion has been debased by rights inflation etc. is an indication that other ways of talking about the issues in question are called for.
Rights talk, however, still has a place, but perhaps a less exalted one than has been fashionable. I suggest that it arises naturally from the implicit rules which govern human interaction. Queuing is a typical example. One has the right to be dealt with before someone who arrives later; they have no right to push in.
In the same way that rights arise from implicit rules of behavior, so, of course, they arise from explicit laws, regulations and legally-enforceable charters or bills of rights. The extent to which rights-specifying laws are well-grounded and respected are important issues, as is the possibility of unintended consequences, such as the rights assigned to one group infringing on the rights of another, or, in the case of bills of rights, the possibility that they may undermine rights that existed under the pre-existing legal status quo.
In general, negative rights (rights not to be interfered with) are less controversial than positive or benefit rights (rights to some good, e.g. food or education). The burdens on other members of the community are less onerous in the former case, though someone's right always entails someone else's obligation, as well as, arguably, (future) obligations on the holder of the right.
One (negative) right which is particularly interesting is the (multi-dimensional) right to privacy. Though culturally relative - there are huge differences in notions of privacy in different human societies - it is one right that is clearly applicable only to self-aware beings. Nobody advocates granting this right - so far as I know - to animals, and it applies only in a tenuous sense to human infants.