The explanation I learned when I first learned the term Poincare patch on AdS$_{d+1}$ in the context of AdS/CFT is the following. The metric for such a patch is
\begin{equation*}
ds^2 = \frac 1{z^2} \left(dz^2 + \eta_{\mu\nu} dx^\mu dx^\nu \right)
\end{equation*}
where $x^\mu$ are coordinates on a $d$-dimensional Minkowski space and $\eta_{\mu \nu}$ is the Minkowski metric, and $z > 0$. An obvious family of isometries of this metric are those which leave $z$ fixed and transform the $x^\mu$ by an isometry of Minkowski space. Of course, these isometries are, as a group, known as the Poincaré group. While we can't necessarily claim that the Poincaré group is a subgroup of the isometry group of AdS, we do certainly get containment at the level of algebras; that is, AdS has a family of Killing fields with the algebra $\mathbf R^{d-1,1} \rtimes \mathfrak{so}(d-1,1)$.
Among physicists, this explanation makes sense from a modern perspective. These isometries exactly give the the Poincaré symmetry on the boundary CFT. Hence, if one wants to break the conformal symmetry while maintaining Poincare symmmetry (as in many applications such as RS models and holographic QCD) the AdS picture is clear. There's no doubt in my mind that the term "Poincaré patch" is so prevalent is at least partially due to this convenience; otherwise it would seem like "AdS-Rindler coordinates" or something similar would fit just as well.
However, this is not the origin of the term. Poincaré is involved more directly. In a series of works on hyperbolic space (beginning with [1]), Poincaré found a Riemannian metric (now called the Poincaré metric) with constant curvature $-1$ on the upper half-space, given by $ds^2 = (dx_1^2 + \cdots + dx_{n-1}^2 + dx_n^2)/x_n^2$. This is most famously known in dimension 2 (where it is related to complex analysis), but easily generalizes to higher dimensions.
The first application of Poincaré's ideas to de Sitter space which I could find is a rather short note: [2]. The result of this work is showing that the Lorentzian version $ds^2 = (dx_1^2 + \cdots + dx_{n-1}^2 - dx_n^2)/x_n^2$ can be isometrically embedded in de Sitter space, which provides a geodesic completion. While the term "Poincaré patch" doesn't appear explicitly anywhere in this paper, it seems to be a simple leap in terminology from "Lorentz-Poincaré metric on a patch of dS" to "Poincaré patch of dS", and then analogously to AdS.
The earliest paper I could locate using the term "Poincaré patch" is the rather famous paper [3], which gives no citation for it nor an explanation of the meaning, but I'm relatively confident it comes from the Lorentzian version of Poincaré's work on hyperbolic space as described above.
References:
[1] Henri Poincaré (1882) "Théorie des Groupes Fuchsiens", Acta Mathematica.
[2] NOMIZU, Katsumi. The Lorentz-Poincaré metric on the upper half-space and its extension. Hokkaido Math. J. 11 (1982), no. 3, 253--261. doi:10.14492/hokmj/1381757803. http://projecteuclid.org/euclid.hokmj/1381757803.
[3] Bañados, M., Henneaux, M., Teitelboim, C., Zanelli, J.: Geometry of the 2+1 black hole. gr-qc/9302012; Phys. Rev. D48, 1506 (1993)
