The minimizer $m$ is the nearest point projection of $X$ onto the subspace of $L^p$ formed by the constant functions ($p=3$ in your case). This $m$ is sometimes called the $p$-prediction or $p$-predictor of $X$. Apparently, this terminology began with Andô and Amemiya. Some of later papers are Landers and Rogge (who wrote a few other papers, e.g. this one), and Cuesta and Matrán. The term "generalized (conditional) expectation" also appeared.