In what follows I will show that the closure of $\ell^1$ under the norm $|x|=\int_1^2|x|_pdp$ is nothing but the [Orlicz space][2] $L_\Phi$, where $\Phi$ is the function
$$\Phi(t)=\frac{t^2-|t|}{\ln|t|}$$
(extended by continuity to $\Phi(0)=0$).
More precisely, I will prove the following claim.

**Claim.** Let $x\in\bigcap_{p>1}\ell^p(\mathbb N)$. Then,
$$ \int_1^2|x|_pdp<\infty \iff \int_1^2|x|^p_pdp<\infty \iff \sum_{j\in\mathbb N} \Phi(x_j)<\infty. $$

**Remarks.**
1) The function $\Phi$ [has a nice graph][1] (it is convex, the derivative is $\Phi'(t)=\frac{t}{(\ln t)^2}$).
2) The function $\Phi$ can be modified to behave essentially arbitrarily at infinity, since functions with the above conditions are bounded. One can take any Young function that behaves like $-|t|/\ln|t|$ as $t\to 0$ for the definition of the Orlicz space.
3) The hypothesis $x\in\bigcap_{p>1}\ell^p(\mathbb N)$ is just for context, one simply needs $x\in \mathbb R^{\mathbb N}$ (with the convention that the $\ell^p-$norm of a sequence not belonging to $\ell^p$ is infinite).
4) The proof is immediate for the zero sequence, so I will assume that $x\not\equiv 0$.

We now begin to prove the claim.
___
**Lemma 1.** Let $f:(0,1]\to(0,+\infty)$ a continuous non-increasing function on $[0,1]$. Then,
$$ \int_0^1 f(x)dx<\infty \implies \sup_{x\in(0,1]} xf(x)<+\infty. $$

*Proof (just an idea; this works only for $C^1$ functions)* One has
$$C:=\int_0^1 f(x)dx=-\int_0^1 xf'(x)dx.$$
Then,
$f(x)=-\int_x^1f'(y)dy\leq \frac{1}{x}\int_0^1yf'(y)dy\leq C/x$.
___
**Lemma 2.** Let $f:(0,1]\to(0,+\infty)$ a continuous non-increasing function on $[0,1]$ s.t. $f(x)\leq C/x$ for some $C>0$. Then,
$$ \int_0^1 f(x)dx<\infty\iff \int_0^1 |f(x)|^{x+1}dx<\infty.$$

*Proof.* It comes from the fact that
$$ \lim_{x\to 0^+} \frac{|f(x)|^{x+1}}{f(x)}=\lim_{x\to 0^+} |f(x)|^x=1,$$
where the last limit follows by monotonicity as $\lim_{x\to 0^+} 1/x^x=1$ and $\lim_{x\to 0^+} c^x=1$, $c>0$.
___
**Proposition.** The first equivalence of the claim holds.

*Proof.* The direction '$\implies$' follows combining the two Lemmas. The other direction follows by monotonicity.
___
The remaining part of the claim follows by Fubini's theorem:
$$ \int_1^2|x|^p_pdp=\int_1^2\sum_{j\in\mathbb N}|x_j|^pdp=\sum_{j\in\mathbb N}\int_1^2|x_j|^pdp= $$
$$ =\sum_{j\in\mathbb N}\frac{|x_j|^2-|x_j|}{\ln|x_j|}. $$
___
**Final remarks.** The claim (together with the observations from other answers) tells that the completion of $\ell^1$ under this norm is the space of the sequences $x$ such that
$$ \sum_{j\in\mathbb N}\Phi(x_j)<\infty. $$
Since this condition is invariant under multiplying $x$ by a non-negative scalar, the space is precisely the [Orlicz space][2] $L_\Phi$. It follows from general facts that the norm $|\cdot|$ is equivalent to
$$ \|x\|_\Phi:=\inf\left\{k\in(0,\infty)\;\Big|\;\sum_{j\in\mathbb N}\Phi\left(\frac{x_j}{k}\right)\leq 1\right\}. $$
I am not sure whether there is a standard way of denoting the space $L_\Phi$ (maybe $\frac{\ell}{\log\ell}$).


  [1]: https://www.wolframalpha.com/input?i=plot%20%28x%5E2-x%29%2Fln%28x%29
  [2]: https://en.wikipedia.org/wiki/Orlicz_space