Here’s a question that Yoav Moriah has suggested. A Heegaard splitting of a 3-manifold with torus boundary (say, a knot complement) is called PADed if it has the following: (1) A planar surface P properly embedded in one of the compression bodies so that all but exactly one of the boundary loops are in the torus boundary of the 3-manifold. (2) An anulus A properly embedded in the same compression body with exactly one boundary loop in the boundary of the 3-manifold. The intersection of A and P must be a collection of arcs that are essential in both surfaces. (3) A disk D in the opposite compression body (or handlebody) such that the boundary of D is disjoint from the boundary of A and intersects P in a single point.
If we Dehn fill the 3-manifold along the boundary slope defined by P then the planar surface P extends to a disk in the resulting 3-manifold. This disk intersects the disk D in a single point so the image of the Heegaard surface is stabilized after the filling. Back in the original 3-manifold, we can spin P around the annulus A generating a new planar surface with a new boundary slope, but that still intersects D in a single point. Filling along this new boundary slope again produces a 3-manifold in which the image of the Heegaard splitting is stabilized. Since we can spin P around A repeatedly, there are infinitely many surgery slopes in which the Heegaard splitting is stabilized. Moriah asks whether the converse is true: Given an irreducible Heegaard splitting of a 3-manifold with a torus boundary such that the Heegaard splitting is stabilized for infinitely many Dehn fillings, is the Heegaard splitting necessarily PADed?
There is a slightly weaker question that one can ask: If the Heegaard genus of a 3-manifold drops for infinitely many Dehn fillings of a boundary torus, does the 3-manifold necessarily have a minimal genus, PADed Heegaard splitting? Yoav Rieck has pointed out that the answer to this question is YES. The answer follows from the proof (though not from the result) in Rieck and Sedgwick’s papers  and  on Heegaard genus after Dehn filling. They show that given a 3-manifold with torus boundary, there are finitely many “lines” in the space of Dehn fillings where the Heegaard genus drops by one, plus a finite number of slopes where it drops by one or more. Although it’s not stated in their Theorem and they don’t use this terminology in the paper, the proof suggests that the lines correspond to PADed Heegaard splittings of the original 3-manifold.
 Finiteness results for Heegaard surfaces in surgered manifolds. Comm. Anal. Geom. 9 (2001), no. 2, 351–367.
 Persistence of Heegaard structures under Dehn filling. Topology Appl. 109 (2001), no. 1, 41–53.