Is there a name for this "mean"?

The quantity $XM$ lies between the arithmetic and geometric means, that is $$AM\geq XM\geq GM.$$ Notice that $$3AM^2=2XM^2+QM^2,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ and so since $AM\leq QM$, it follows that $$3AM^2=2XM^2+QM^2\geq 2XM^2 +AM^2\Rightarrow AM\geq XM.$$ The AM-GM inequality implies that $XM\geq GM$. From $(1)$ we may write $$XM=\sqrt{\frac{3AM^2-QM^2}{2}},$$ but a nicer way to express $XM$ is given in Thomas Andrews comment: $$XM=GM\sqrt{\frac{GM}{HM}}.$$

See Also: Newton's inequalities. More generally quantities such as $XM$ are often referred to as Elementary Symmetric Means.