If $X$ is a set containing element $0$, and we want the set that contains all elements in $X$ except $0$, we can write $X-\{0\}$.
If $X=\prod_{i=1}^n X_i$ and we want to denote the cartesian product $Y$ of all $X_i$, except some $X_j$, can we similarly write $Y=X/\{X_j\}$?
This seems wrong, because $X_j$ is not a subset of $X$. How do we write this?
Similarly, if we have an element $x\in X$, but we want the element $y\in Y$ that is a tuple equal to $x$, except with the $j^{th}$ element removed, (i.e. the element in $X_j$), then how do we write this?
(I think it's called the "projection" of $x$ onto $Y$? Is that true?)
$X\setminus\{0\}$ (or $X-\{0\}$) is commonly used as notation for the set that contains all elements of $X$ except $0$ (not $X/\{0\}$ as you suggest, which has the looks of a quotient).
You can write:$$Y=\prod_{i=1,i\neq j}^nX_i$$ to denote the product of all $X_i$ except $X_j$.
I suspect that $x=(x_1,\dots,x_n)$ and you want a notation of $(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)$. This is a notation on its own already. Sometimes it is denoted as $$\left(x_{1},\dots,\hat{x_{j}},\dots,x_{n}\right)$$but if you do that then feel obliged to inform the reader.
Formally $X=\prod_{i=1}^nX_i$ is a product equipped with projections $p_i:X\to X_i$ for every $i$ prescribed by $(x_1,\dots,x_n)\to x_i$, so label "projection" is already preoccupied in some sence. On the other hand $X$ can also be looked at as a product $Y\times X_j$ having projections $X\to Y$ and $X\to X_j$. In that context the function $X\to Y$ that leaves $x_j$ can be recognized as a projection.