Multiple mirrors phenomenon from SYZ and HMS perspective There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties but let's forget about them for now. 
There are several more or less precise incarnations of mirror symmetry, including so-called SYZ and HMS programs. Each gives us a certain recipe to construct the manifold mirror to a given Calabi--Yau. 
In SYZ, the expectation is that to construct the mirror manifold we need to put our Calabi--Yau in a family of Calabi--Yau's over the punctured disc so that the monodromy operator is maximally unipotent. Different maximally unipotent degenerations can lead to different mirror manifolds. 
In HMS, let's assume that we have finally found the right definition of Fukaya category. After certain algebraic manipulations, we can cook up a triangulated category which should be equivalent to the derived category of coherent sheaves on the mirror manifold. If we pick a t-structure in that category, we can Gabriel-style reconstruct the mirror manifold itself from the heart of t-structure. However, if I understand correctly, there is no natural choice of t-structure so we may get several mirror manifolds. 
The question is: are these two ambiguities (choosing a maximally unipotent degeneration and choosing a heart of t-structure) in some sort of bijective correspondence? Are there some references where this is studied? 
P.S.: to avoid certain subtleties in hyperkaehler geometry, let's assume that the holonomy group of Calabi--Yau's is equal the special unitary group. 
EDIT: on second thought, it feels like this should be wrong (Calabi--Yau's can be derived equivalent for reasons that have nothing to do with mirror symmetry). Is there some way to single out the mirrors arising from SYZ inside the set of mirrors arising from HMS? 
 A: Some relation between these two ambiguities is indeed part of the expected but still largely conjectural big picture.
Let $X$ be a compact Calabi-Yau manifold, viewed in a symplectic way. Let $M_X$ be the moduli
space of complex structures on $X$. The various maximally unipotent degenerations are the various ways to go
to infinity in $M_X$ in a maximally degenerated way. Some picture useful to have in mind for $M_X$ in some non-compact 
object, with various "cusps" corresponding to the various maximally unipotent degenerations.
Let $F(X)$ be the Fukaya category of $X$. It is expected that each choice of complex structure on $X$, 
i.e. of point of $M_X$, defines a stability condition on $F(X)$ in the sense of Bridgeland.
A stability condition on $F(X)$ includes the data of a t-structure on $X$. The expectation is that this 
stability condition should be constructed in terms of the geometry of special Lagrangians in $X$ (the notion of
special Lagrangian depends on the holomorphic volume on $X$ determined by the chosen complex structure on $X$),
see for example https://arxiv.org/abs/1401.4949
For each choice $j$ of maximally unipotent degeneration, i.e. each "cusp" of $M_X$, 
we expect to have a SYZ fibration from which it is possible to construct a mirror $Y_j$, with some derived
category of coherent sheaves $D(Y_j)$ equivalent to $F(X)$. The general expectation is that, if we consider the
family of stability conditions defined by a family of complex structure going to $j$, then the corresponding $t$-structures
have for limit the standard $t$-structure on $D(Y_j)$ whose heart is the abelian category of coherent sheaves on $Y_j$.
More precisely, one should be able to identifiy a neighborhood of the cusp $j$ in $M_X$ with an open part of the Kähler cone of 
$Y_j$ and the limit "go to j" should be the "large volume limit" for $Y_j$. A semi-expository version of this story can be found in the book
http://www.claymath.org/library/monographs/cmim04.pdf
Some more technical point: one should rather consider the universal cover of $M_X$, the fundamental group acting on $F(X)$ by autoequivalence,
or equivalently stability conditions up to autoequivalences.
A remark about your "second thought": I think that all the known examples of derived equivalences between Calabi-Yau manifolds
have something to do with mirror symmetry. There are no known a priori reasons why it should be the case, but I don't know a clear 
counterexample either.
A: Two algebraic varieties are called Fourier-Mukai partners if their bounded derived categories of
coherent sheaves are equivalent.
A "trivial" example of Fourier-Mukai partners is given by birational Calabi-Yau manifolds (it is a conjecture proved only for dimension 3 by Bridgeland, for holomorphic symplectic varieties by Kaledin, and in few other cases).
Modulo birational equivalence all other Fourier-Mukai partnerships are expected to be related to mirror symmetry. Usually in these examples there is also some "Homological Projective Duality" phenomenon or an explanation in terms of a variation of stability condition for GIT quotient /  also referred as Fayet--Iliopoulos parameter of some Gauged Linear Sigma Model.
The first example of this phenomenon for CY3 is the so-called Pfaffian/Grassmannian correspondence between Calabi--Yau threefolds obtained as linear sections of the Grassmannian variety Gr(2,7) of lines in $\mathbf{P}^6$ and its projective dual Pffaffian variety Pf(4,7) of skew-symmetric two-forms in 7 variables of rank at most 4. The double-mirror phenomenon was observed by Rodland and van Straten, derived equivalences are proved by Borisov--Caldararu and Kuznetsov, also Segal et al. give a proof in spirit of GLSM argument of Hori--Tong.
Another ingeresting  example is in a series of papers by Hosono--Takagi on double quintic symmetroids and Reye congruences. A feature of this example is that one of CY3s is simply-connected while another is not. As far as I remember they also review the expectations similar to your question and my answer in some detail.
Roughly  mirror conjectures imply that the subgroupoid of the fundamental groupoid of the moduli space of complex structures of mirror dual Calabi--Yau manifolds whose objects are Maximally Unipotent Monodromy points is equivalent to the groupoid of the derived equivalences between Fourier-Mukai partners of the original Calabi-Yau manifold. This statement also includes a bit better known conjecture of Kontsevich that the group of derived autoequivalences corresponds to the monodromy group of the mirror family. In physics literature there are some nice studies of Knapp and Romo.
