Norbert Wiener, in his book Ex-Prodigy, quite clearly testifies to Sidis' brilliance in mathematics at a young age. That is the single most powerful bit of evidence in favor of the premise that Sidis was the real deal. Wiener was not close to him, but knew him at Harvard, had a class with him, and had some conversations with him. He also knew him later in life, when William was haunting the halls of MIT for employment doing computational work, of which there was plenty in the 1940's - but no more, because he did not want to get too deeply into mathematics any longer. Sparse contact, but real, not hearsay.
The short answer is that Wiener hints that Sidis' non-mathematical accomplishments, his languages and varied subjects of curiosity, are padded and his physics was suspect, but his mathematical potential was top-shelf. He also confirms some rumors of Billy's eccentricities and contradicts others. As Wiener was himself top-shelf, I take that seriously enough to accept the evaluation with no further evidence. Had NW said that about a laboratory custodian at MIT with no other credentials I would accept the judgement. That excellence in math, in turn, gives at least some support that the other precocities might be somewhat true. We don't have to believe that he taught himself Latin and Greek at 4, as his mother claimed, to believe that he was very good.
Yet these find no echo in his later life. The mathematics, though he came to avoid it, at least shows up off and on. The languages, not at all. Not even a passing mention by friends, family, interviewers, reporters later - only the languages that he supposedly knew when young. We might conclude that he had ability but no further interest, and let it slide, or that he kept the abilities only for personal use or amusement.He does not seem to have read widely in other languages, at any rate.
Language prodigies tend not to pan out. Languages as read and as spoken are different, and the subtleties of expression somehow do not register with them as well. In Sidis's case, social subtleties eluded him even in English, so it would hardly be surprising that nuance eluded him in other tongues. Math and music prodigies are most common, languages after that, and males far more common than females. They translate the literal sense of things, they know rules and constructions. They do not get called on to translate literature or diplomatic speech - those arts seem to develop slowly.
Still, Sidis might have known enough of many languages to understand much of what is written in them. What number we claim he "knew" - five or forty or a hundred - is likely to depend strongly on what we call "knowing." There was no fluency in Pennacook, as there were no native speakers but there might have been enough to cobble together a meaning from any document that came to light. A useful skill, and one that demonstrates high intelligence. But "genius" would seem to require more than mere accumulation.
The categorising of streetcar transfers makes immediate sense to anyone who understands the OCD or autism spectra. It gives pleasure, and there is no reason not to. But spinning lengthy theories about Atlantis, Red Men, and the Constitution with no real grounding, how do we rank that? Do we simply award no points in our estimate of his intelligence, or do we take them away? Similarly with philosophy stretched over physics - originality is a good thing, but what are we to make of a person who cannot look at his work and say "Wait, this just doesn't add up?"
In a rough chronological order, William James held a theory about reserve energy. Boris Sidis learned of reserve energy from William James. Boris claimed that William’s intelligence was in part due to the use of reserve energy. William Sidis pondered how William James’ theory of reserve energy might be removed from the metaphysical realm and brought into a scientific study. The Animate and the Inanimate is William Sidis’ pondering about how the second law of thermodynamics might apply to William James’ reserve energy theory…Does that even mean anything?
We can't know what William might have been if he was brought up by a less driven father - or if someone had whacked his father a few times and made him see the obvious. Boris pushed and William broke. Would he have broken anyway? Would he have been brilliant anyway? Not only unknown, but currently unknowable.
Let us assign some modest percentage to the possibility that Norbert Wiener was fooled somehow - that reputation plus William's computational feats plus overidentification because Wiener was himself a prodigy with a driven father caused him to believe that young William was smarter than he was. The greater likelihood remains that Wiener sized him up correctly, and Sidis had it in him to contribute to 20th C mathematics at highest levels. Even at that, even granting everything, I don't see him at the very top. Smarter than anyone you are likely to run into this week, perhaps. But we are entering territory where the competition is stiff as well. There isn't evidence that William James Sidis did anything more than moderately special, he just did it younger. That's not enough. Precocity is not in itself proof.
We imagine that there is an arc to intelligence over the lifespan, and that a steeper trajectory in youth must mean a higher peak. There is some evidence for this. But in music, and athletics, in drawing, and performing, and chess, we know of examples of early promise that simply peaked before their age mates but ultimately rose no higher than the others in the top 1% in the field. Just earlier, not better. If we are going to speculate about some mind that William Sidis would have had if, we have to also take it as it is - ultimately not focused, not able to sort good ideas from bad, original but not precise.
So what is your estimate of his intelligence? One in a hundred? One in a thousand? A million? A billion?