I don't know the real answer to your question, anyway I'll try to give to you a possibile explanation, following the suggestion of Henning Makholm.
Bifunctors can be consider a categorical version of bilinear map. A bilinear map is just a map $f \colon V \times W \to Z$, where $V$, $W$ and $Z$ are vector spaces considered just as sets, which is linear in each component. In a similar way a bifunctor is map $\mathcal F \colon \mathbf C \times \mathbf D \to \mathbf E$ of graphs, which is functorial in each component (but it has also to satisfy a condition). Before I can justify my statement let me state the following theorem:
Suppose you have two families of functors $\langle L_c \colon \mathbf D \to \mathbf E\rangle_{c \in \mathbf C}$ and $\langle R_d \colon \mathbf C \to \mathbf E\rangle_{d \in \mathbf D}$ such that:
for each pair $c \in \mathbf C$ and $d \in \mathbf D$ we have that $L_c(d)=R_d(c)$
for each pair $f \in \mathbf C(c,c')$ and $g \in \mathbf D(d,d')$ the equality $L_{c'}(g) \circ R_d(f) = R_{d'}(f) \circ L_c(g)$ holds
then there exists a necessarily unique functor $\mathcal F \colon \mathbf C \times \mathbf D \to \mathbf E$ such that $\mathcal F(c,d)=L_d(c)=R_c(d)$ and $\mathcal F(f,g)=L_{c'}(g)\circ R_d(f)=R_{d'}(f)\circ L_c(g)$.
So given a graph map $\mathcal F \colon \mathbf C \times \mathbf D \to \mathbf E$ such that for each $c \in \mathbf C$ and $d \in \mathbf D$ the components $\mathcal F(c,-)$ and $\mathcal F(-,d)$ are functors (which have to satisfy the condition of the theorem above), then $\mathcal F$ is a bifunctor, and $\mathcal F$ can be a bifunctor if and only if it satisfies this condition.
Anyway there's more: a bilinear map $f \colon V \times W \to Z$ (or reather $f \colon V \otimes W \to Z$) can be seen both as a map $f \colon V \to \hom(W,Z)$ and $f \colon W \to \hom(V,Z)$ through the natural bijections
$\hom(V,\hom(W,Z)) \cong \hom(V \otimes W,Z) \cong \hom(W,\hom(V,Z))$
and this is the real cool property of bilinear maps in the category of vector spaces.
In the category of (small) categories and functors instead we have these natural bijections
$\hom(\mathbf C, \hom(\mathbf D,\mathbf E)) \cong \hom(\mathbf C \times \mathbf D, \mathbf E) \cong \hom(\mathbf D, \hom(\mathbf C,\mathbf E))$
so in categorical context the role played by the tensor product for vector spaces is played by product and thus the role of bilinear maps is played by bifunctors.
I hope this explanation can be considered enough reasonable to justify the name bifunctor.