Let assume $Y$ is a closed oriented rational homology sphere.

The set of spin$^c$ structure of $Y$ is in one to one correspondence with $H_1(Y,\mathbb{Z})$. So if $H_1(Y,\mathbb{Z}) \neq 0$ then there are more than one spin$^c$ structure.

The question then becomes: When does $\widehat{HF}(Y,\mathfrak{s}) \cong \mathbb{Z}$ for each spin$^c$ structure $\mathfrak{s}$?

Three-manifolds with such property are called $L$-spaces. Double branched cover of quasi-alternating knots (in $S^3$) are L-spaces. I think your question is a work in progress. For some other examples see Steven Sivek comment below.

Note also that the collection of quasi-alternating knots contains the collection of alternating knots, and 2-bridge knots are examples of alternating knots.

In the special case $H_1(Y,\mathbb{Z}) = 0$, in which situation there is a unique spin$^c$ structure, it is conjectured that $S^3$ and the Poincar\'e homology sphere ($\Sigma=S^3/I$ where $I$ is the full icosahedral group) are the only such 3-manifold. We know that $d(S^3)=0$ and that $d(\Sigma)=\pm 2$ depending on the orientation you chose. For the general case I think knowing exactly the $d$-invariant is still an open problem except when $Y$ is a lens space in which case there is a formula for each $d$ invariant.