gn.general topology - Is this a known compactification of the natural numbers?

I was about to answer the same thing as Joel, but here is a more topological perspective. I will show that the lattice of open subsets of $overline{mathbb{N}}$ is very similar to the lattice of open subsets of the Stone-Čech compactification $betamathbb{N}$. So, from the localic point of view, there is but a small difference between your space $overline{mathbb{N}}$ and $betamathbb{N}$.

View $betamathbb{N}$ as the set of ultrafilters on $mathbb{N}$ and let $mathbb{N}^*$ be the remainder $betamathbb{N}setminusmathbb{N}$. (Prinicipal ultrafilters are identified with the corresponding point of $mathbb{N}$, so $mathbb{N}^*$ is the subspace of nonprincipal ultrafilters.) Recall that the clopen subsets of $betamathbb{N}$ are precisely those of the form

$$langle A rangle = { mathcal{U} in betamathbb{N} : A in mathcal{U} }$$

for $A subseteq mathbb{N}$. Note that $A preceq B$ iff $langle A rangle capmathbb{N}^* subseteq langle B ranglecapmathbb{N}^*$, so the points of $partialmathbb{N}$ can be identified with the clopen subsets $[A] = langle A rangle cap mathbb{N}^*$ of the remainder $mathbb{N}^*$ (including the empty set). Given an open set $U subseteq betamathbb{N}$, the set

$$U' = (U cap mathbb{N}) cup { [A] : [A] subseteq U }$$

is open in $overline{mathbb{N}}$. Conversely, given an open set $V subseteq overline{mathbb{N}}$, your conditions ensure that

$$V' = (V cap mathbb{N}) cup bigcup { [A] : [A] in V }$$

is open in $betamathbb{N}$. This correspondence is not perfect since $A cup B cup [A] cup [B] = A cup B cup [A cup B]$, but

$$A cup B cup {[C] : C preceq A lor C preceq B} quadmbox{and}quad A cup B cup {[C] : C preceq A cup B}$$

are not always the same. However, these are the only errors that occur, i.e. the translation is perfect for open subsets of $overline{mathbb{N}}$ whose part in $partialmathbb{N}$ is upward directed in the ${preceq}$ ordering.

Although your space $overline{mathbb{N}}$ is interesting, this approximate translation suggests most of its applications could be transferred to work over the well studied space $betamathbb{N}$ instead.

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Here is yet another perspective which suggests that there may be more to $overline{mathbb{N}}$ after all. The soberification of $partialmathbb{N}$, which I will denote $mathrm{Fil}_{mathbb{N}}$, is the space of all nonprincipal filters on $mathbb{N}$, with the topology generated by the basic open sets

$$[A] = { mathcal{F} in mathrm{Fil}_{mathbb{N}} : A in mathcal{F} }.$$

The points of $partialmathbb{N}$ can be identified with the filters $mathcal{F}_A = { B : A preceq B }$. Note that the space $mathbb{N}^*$ is also a subspace of $mathrm{Fil}_{mathbb{N}}$, which explains the connection found above.

For the pointless topology aficcionados, the space $mathrm{Fil}_{mathbb{N}}$ is obtained by imposing the trivial Grothendieck topology on the preorder $(mathcal{P}mathbb{N},{preceq})$ viewed as a category (or, equivalently, the quotient partial order $mathcal{P}mathbb{N}/mathrm{fin}$ as suggested by Joel). The subspace $mathbb{N}^*$ is similarly obtained by imposing the finite cover (aka coherent) Grothendieck topology on $(mathcal{P}mathbb{N},{preceq})$.