I don't know a name for it, so let me call "even bipartition principle" the statement that every infinite set $X$ is equinumerous to $X \times \mathbf{2}$, where $\mathbf{2} = \{0, 1\}$ — in other words, every infinite set $X$ can be partitioned into two subsets $Y, Z$ such that $X, Y, Z$ are all equinumerous.
It is pretty intuitive, and quite easily provable in ZFC. For example, one can well-order $X$, then, for all limit ordinal $\kappa$ and all integer $e$, put the element associated to $\kappa + e$ in either part of the partition depending on the parity of $e$. Alternatively, apply Zorn's lemma to the set of functions $f : X \to \mathbf{2}$ such that $|f^{-1}(0)| = |f^{-1}(1)|$, ordered by graph inclusion, then send the one possibly remaining element indifferently to 0 or 1.
On the other hand, as I learn from Noah Schweber here, it is consistent with ZF that there exists an amorphous set, i.e., an infinite set which cannot even be partitioned into two infinite subsets (not even requiring that they are equinumerous to the original set).
A related principle is: every infinite set $X$ can be partitioned into $|X|$ many subsets which are equinumerous to it, or put differently, $|X| = |X \times X|$. This is also provable in ZFC, and it is a theorem of Tarski that it implies choice over ZF. The proof is the following. Let $X$ be an infinite set, and let us prove that $X$ can be well-ordered. Using the Hartogs number construction, we can find (in ZF) an ordinal $\kappa$ which cannot be injected into $X$. By assumption, there is a bijection $f : (X \uplus \kappa) \leftrightarrow (X \uplus \kappa) \times (X \uplus \kappa)$. The image of $X$ cannot cover any of the "fibers" $(X \uplus \kappa) \times \{x\}$ for $x \in X$ because that would give an injection $\kappa \to X$. So, if we define $g(\gamma)$ for $\gamma \in \kappa$ to be the second component of $f(\gamma)$ when it is in $X$, we get a partial function $g : \kappa \dashrightarrow X$ which is surjective. Thanks to the well-ordering on $\kappa$, we can find a section for this surjection without the axiom of choice, by setting $h(x) = \min g^{-1}(\{x\})$, which gives an injection $h : X \to \kappa$. Now define $(x ≤ y) := (h(x) ≤ h(y))$: it is a well-ordering on $X$ by injectivity.
Question: Can we assess the axiomatic strength of this "even bipartition principle"? I don't see a way to modify the proof above in order to prove that it implies choice over ZF. Is it strictly weaker? Is it equivalent to some known variant of choice or any more well-known axiom?
