According to this Wikipedia page, Mendeleev originally believed that the inert gases belonged in Group 0 (to the left of the alkali metals). Thus, helium would be placed in the second period to the left of lithium and argon would be placed in the third period to the left of sodium. Apparently this made sense to him at the time, since atomic theory was in its infancy (and Mendeleev didn't like it) and quantum mechanics was only a twinkle in the eyes of Planck, Bohr, and others.
Based on this arrangement, there is an open spot above helium and to the left of hydrogen for an ultralight inert gas he called "proto-helium". We now know that hydrogen must be the lightest known element, meaning Mendeleev was wrong about the placement of the inert gases. However, host of the other holes in Mendeleev's table were filled with real elements having properties nearly identical to what Mendeleev predicted.
Mendeleev predicted that proto-helium would have a miniscule mass (ca. $ 5.3\times10^{-11}$ amu) and high (near relativistic) velocity. Mendeleev believed that this near massless particle would be able to permeate all matter, rarely interacting chemically. He thought that they might be responsible for radioactive decay.
A neutrino is massless (or nearly so), and travels at the speed of light (if massless) or just under (if it has mass). Otherwise, it ignores the electromagnetic force, and so has no chemical interaction. Neutrinos are often produced during radioactive decay.
The relevant references point back to Mendeleev's original writings (in Russian), which I cannot access, so I cannot verify if he made a prediction about "proto-helium". Did he predict this particle? Has anyone previously made the connection between Mendeleev's proto-helium and neutrinos?