I assume $L^p$ norm means the operator norm on $\mathbb{R}^N$ with the $\ell_p$ norm.

Then by Riesz-Thorin-Stein interpolation, you have $$ \|H\|_p \leq \|H\|_1^{1/p} \|H\|_\infty^{1-1/p} = \sqrt[p]{N} $$ By testing on the vector $(1,0,0,\ldots,0) \mapsto (-1,1,-1,1,\ldots)$ you have $$ \|H\|_p \geq \sqrt[p]{N} $$

and hence $\sqrt[p]{N}$ is the value.

Important Edit: As J.J Green pointed out, the OP contains an incorrectly stated value for $\|H\|_{\infty}$, which I copied without checking below. Interpolating between $(1,\infty)$ using the corrected version would give the trivial bound $\|H\|_p \leq N$. You regain the sharp bound by interpolating instead between $(1,2)$ and $(2,\infty)$.

I assume $L^p$ norm means the operator norm on $\mathbb{R}^N$ with the $\ell_p$ norm.

Then by Riesz-Thorin-Stein interpolation, you have $$ \|H\|_p \leq \|H\|_1^{1/p} \|H\|_\infty^{1-1/p} = \sqrt[p]{N} $$ By testing on the vector $(1,0,0,\ldots,0) \mapsto (-1,1,-1,1,\ldots)$ you have $$ \|H\|_p \geq \sqrt[p]{N} $$

and hence $\sqrt[p]{N}$ is the value.