Solar neutrino deficit was first observed in the late 1960's. And theory of neutrino oscillation was developed in 1967. But,in 2001, the first convincing evidence of solar neutrino oscillation came in SNO. Why did it take nearly 35 years to verify the neutrino oscillation?
The unique thing about SNO was that it was simultaneously sensitive to charged-current and neutral-current interactions, because they used deuterated water.
The three main interactions are
Neutrino capture on deuterium, $\nu + n \to e + p$, which generates a fast electron and a slow proton. The lepton and the baryon exchange a $W$ boson (the "charged weak current"). Only electron-type neutrinos may participate in this interaction; $\nu_\mu$ and $\nu_\tau$ would have to generate heavier leptons, but solar neutrinos don't carry enough energy to make those more massive particles.
Deuterium dissociation due to neutrino scattering, $\nu + (np) \to \nu + n + p$. The free neutron will wander around for a while before getting captured on another deuteron and emitting a gamma ray. Because the neutrino's charge doesn't change this reaction is mediated by the "neutral current" (the $Z$ boson) and all neutrinos contribute equally.
Elastic scattering from electrons, $\nu + e \to \nu + e$. This interaction has both charged- and neutral-current contributions, so neutrinos of all flavors may contribute, but electron neutrinos contribute more heavily than the other flavors.
These different interaction channels gave independent measurements of the total neutrino flux and the electron neutrino flux.
It's worth noting that the neutral current had only just been predicted in 1967, and was not discovered until the early 1970s.
For the most part the solar neutrino community believed that there was some misunderstood property of neutrino detection that caused everybody to measure one-third the predicted solar neutrino flux. It took many years before the possibility that the misunderstood bit was a property of the neutrino itself was really taken seriously.
I don't know for certain, but I would expect that the design discussions for SNO began in the early 1990s. There are many technical challenges associated with the detector — not least that they have many tons of heavy water suspended in many tons of light water in a thin, transparent membrane. The heavy water is on loan from the Canadian nuclear power industry; SNO has a hefty insurance policy to pay to replace it if the membrane ruptures and the heavy water mixes with the light water and is ruined.
In my opinion, there are several reasons:
Solar model calculations: One possible explanation was that there were problems with the solar models that probably underestimate some errors.
Complex experiment: This experiment employed a lot of chemistry techniques that are usually very hard to control. Davis did an incredible job, but only with cleaner experiments based on Cherenkov radiation that we were able to completely confirm the results of Homestake.
Theoretical prejudices: The mixing of quarks was discovered well before the mixing of neutrinos and the mixing matrix in this case (CKM) happens to be almost diagonal. On the other hand, in order to explain the deficit of solar neutrinos, we need relatively large mixing angles. Today, we know that the PMNS matrix is very far from diagonal, but at that time this was not something trivial.