(unusual move to answer my own question)

In fact this is duplicate:

* https://math.stackexchange.com/questions/914152/how-to-mark-rational-points-on-a-sphere

and the rational points are of course easy to find

$$ x=\frac{2u}{u^2+v^2+1};y=\frac{2v}{u^2+v^2+1};z=\frac{u^2+v^2-1}{u^2+v^2+1} $$ 

There is even a proof of equidistribution as a result of [Duke](http://www.math.ucla.edu/~wdduke/preprints/rational.pdf)  He shows that:

$$ \frac{\sum_{h(x)\leq T} \psi(x)}{\sum_{h(x)\leq T} \;\;1\;\;} \to \int_{S^2} \psi \, d\mu $$
for any function $\psi: S^2 \to \mathbb{C}$.  There is also general discussion of Bjorn Poonen in genral on [rational points on varieties](http://www-math.mit.edu/~poonen/papers/Qpoints.pdf)

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In light of the comments it's worth noting the Manin-Peyre conjecture which discusses rational points on Fano varieties.  I don't see how anyone can study rational points on a manifold with no explicity formula.  At least, observe that:
$$ X^3 + Y^3 + Z^3 = 3 \, T^3$$
does not satisfy weak approximation yet it's a cubic surface in $\mathbb{P}^3$ and therefore Fano.  

The full definition of Fano is really fancy involving the anti-canonical divisor and ample line bundles, which I doubt is how Fano or del Pezzo discussed them.

And here is more references:

* [Hauteurs et Measures de Tamagawa](https://www-fourier.ujf-grenoble.fr/~peyre/publications/textes/fano.pdf)
* [An overview of Manin's conjecture for del Pezzo surfaces](https://arxiv.org/abs/math/0511041) (many explicit examples)