Following the specific reference provided by Steve Huntsman in his MO answer leads to the paper
This paper presents randomized reductions from the approximation versions of lattice problems in the $\ell_2$ norm to any norm $\ell_p$ where $1 \le p \le \infty$. The reductions don't quite preserve the approximation ratio, but the loss can be bounded by a factor of $(1+\epsilon)$. So the $\ell_2$ norm yields in some sense the "easiest" versions of lattice problems. It is enough to prove hardness of approximation using the $\ell_2$ norm, and hardness of approximation for any other $\ell_p$ norm follows with only a $(1+\epsilon)$ larger approximation factor.
Regev and Rosen then go on to use this to strengthen some existing hardness results.
On the other hand, Chris Peikert showed that lattice problems posed using $\ell_p$ norms are not substantially harder than problems posed in the $\ell_2$ norm.
So yes, the question is sensible, and it has also essentially been answered.
(Thanks to @user834 for the preprint links.)