In Terry's answer, he shows that his original question reduces to the question of whether, given a $\sigma$-algebra $\mathcal F$ on some set $X$ and a measure $\mu$ on $(X,\mathcal F)$, there is a ``splitting'' of the quotient algebra $\mathcal F / \mathcal N$, where $\mathcal N$ denotes the ideal of $\mu$-null sets. In this context, a splitting is a Boolean homomorphism $\Phi: \mathcal F / \mathcal N \rightarrow \mathcal F$ such that $\Phi([A]) \in [A]$ for all $A \in \mathcal F$. (Some authors call this a lifting instead of a splitting.) When some such $\Phi$ exists, let us say that $(X,\mathcal F,\mu)$ has a splitting.
I did some digging on this question this afternoon, and found two very good sources of information: David Fremlin's article in the Handbook of Boolean Algebras (available here) and a survey paper by Maxim Burke entitled "Liftings for noncomplete probability spaces" (available here). I'll summarize some of what I found below to supplement what Terry mentions in his answer. He mentions already that it is independent of ZFC whether $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting:
$\bullet$ (von Neumann, 1931) Assuming $\mathsf{CH}$, $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting.
$\bullet$ (Shelah, 1983) There is a forcing extension in which $([0,1],\text{Borel},\text{Lebesgue})$ has no splitting.
Also mentioned already is the fact that if we expand the $\sigma$-algebra in question from the Borel sets to all Lebesgue-measurable sets, then the situation is more straightforward:
$\bullet$ (Maharam, 1958) If $(X,\mu)$ is a complete probability space, then $(X,\mu\text{-measurable},\mu)$ has a splitting.
Now on to some not-yet-mentioned results. First, it's worth pointing out that one can obtain splittings with nice extra properties.
$\bullet$ (Ioenescu-Tulcea, 1967) Let $G$ be a locally compact group, and let $\mu$ denote its Haar measure. Then $(G,\mu\text{-measurable},\mu)$ has a translation-invariant splitting (which means $\Phi([A+c]) = \Phi([A])+c$ for every $\mu\text{-measurable}$ set $A$).
Once again, shrinking our $\sigma$-algebra from all $\mu$-measurable sets to only the Borel sets causes problems.
$\bullet$ (Johnson, 1980) There is no translation-invariant splitting for $([0,1],\text{Borel},\text{Lebesgue})$.
Thus, interestingly, Shelah's consistency result becomes a theorem of $\mathsf{ZFC}$ if we insist on the splitting being translation-invariant (with respect to mod-$1$ addition). More generally:
$\bullet$ (Talagrand, 1982) If $G$ is a compact Abelian group and $\mu$ is its Haar measure, then there is no translation-invariant splitting for $(G,\text{Borel},\mu)$.
What stood out to me most in Fremlin and Burke's articles is how many questions seem to be wide open.
Open question: Is it consistent that every probability space has a lifting?
If yes, this would give a consistent positive answer to Terry's original question.
Open question: Is it consistent with $2^{\aleph_0} > \aleph_2$ that $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting?
(Carlson showed that it is consistent to have $2^{\aleph_0} = \aleph_2$ and for $([0,1],\text{Borel},\text{Lebesgue})$ to have a splitting. Specifically, he showed that this holds whenever one adds precisely $\aleph_2$ Cohen reals to a model of $\mathsf{CH}$.)
Open question: Does Martin's Axiom (or $\mathsf{PFA}$, or $\mathsf{MM}$) imply that $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting?
Open question: What of the same question in other well-known models of set theory (the random model, Sacks model, Laver model, etc.)?
