I am currently studying certain infinite graphs in terms of their finite induced subgraphs. For the graphs that I am interested in the class of finite induced subgraphs is closed under the following operation:

Given two graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ and a vertex $v$ of $G$, let $G\otimes_v H$ (notation invented on the spot by myself) be the graph on the disjoint union of $V(G)\setminus\{v\}$ and $V(H)$ in which two vertices $x$ and $y$ are connected by an edge iff one of the following holds:

(1) $x,y\in V(G)\setminus\{v\}$ and $\{x,y\}\in E(G)$.

(2) Exactly one of $x$ and $y$ is in $V(H)$, say $y$ (wlog), and $\{x,v\}\in E(G)$.

(3) $\{x,y\}\in E(H)$.

In other words, the vertex $v$ of $G$ is replaced by a copy of $H$, and every
vertex $w$ of $G$ different from $v$
is connected to all vertices of the copy of $H$ if
$w$ and $v$ are connected in $G$. Otherwise $w$ is not connected to any of the vertices of the copy of $H$.

(Note that I am only doing this at a single vertex of $G$, not all of them. Otherwise I would get the wreath product or lexicographic product as mentioned in Nathann Cohen's answer below.)

Since this is a natural operation between graphs (with a distinguished vertex of the first graph), I would guess this has a name. If yes, how is this called?