Let $E_1,...,E_n$ be vector spaces. We know that a function $p_i: \prod_k E_k \to E_i$, $p_i(x_1,...,x_n) = x_i$ is called a projection function.
I often have to use the function $u_i: E_i \to \prod_k E_k$, $u_i(t) = (0,...,0,t,0,...,0)$ (i.e. $t$ at the $i$th position, and $0$ everywhere else).
Is there a common name for such a function? If not, have you ever seen someone using a certain name for it?
This is often called an inclusion (map). The term applies to more than linear maps, and is generally used to describe an injective map $$\iota: X \hookrightarrow Y$$ in situations where we want to regard $X$ as a particular subset of $Y$ by identifying $X$ with $\iota(X)$ via $\iota$.
For spaces $X_i$ with preferred respective points $a_i \in X_i$, we get canonical (natural) inclusions $\iota_i: X_i \hookrightarrow \prod_j X_j$ defined by $$\iota_i(x) := (a_1, \ldots, a_{i - 1}, x, a_{i + 1}, \ldots), $$ and the inclusions $u_i$ in the question are just the special case of these for which $X_i = E_i$ and the $a_i$ are the respective zero elements of $E_i$. By construction, the canonical inclusion $\iota_i$ is a right inverse for the canonical projection $\pi_i : \prod_j X_j \to X_i$ defined by $\pi_i(x_1, x_2, \ldots) = x_i$, that is, $\pi_i \circ \iota_i = \operatorname{id}_{X_i}$.