The anions of the form $\ce{MnO_x^y-}$ are referred to as manganates (see Wikipedia). I'm not sure if there might be a "special" name for $\ce{MnO2^-}$ specifically (that's the species you have here) because I never encountered this anion in my lab times, but given that $\ce{MnO4^2-}$ are the "normal" manganates and $\ce{MnO4^-}$ the permanganates, I think manganate would be the term used here as well.
Now to distinguish the ambiguity in $x$ and $y$ you usually add the oxidation state of the central metal (in this case manganese) in the form of roman numbers. Since actually the same stoichiometric formula ($\ce{NaMnO2}$) could give rise to different stable crystal structures (depending on external conditions) one uses the greek letter to distinguish between these different structures. So, yes, to be precise the $\alpha$ is needed and I would call the compound $\alpha$-Sodium-manganate(III) pronounced "alpha sodium manganate three".
But in day to day conversations one would rarely use the full thing if it is clear from context. First thing usually the $\alpha$ is dropped and next the (III). So taking to a lab colleague, where it is clear you can just say Sodium manganate.
Update: I just saw that the materialsproject has something on your compound under ID mp-18957, where they tagged it as Sodium manganese(III) oxide
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