Increase the probability that homeopathy works for me- but also increase the probability of all other competing hypotheses except the "sugar placebo" one. The next most plausible hypothesis at that point is probably that the effort is related to my mother's recommendation. Unless I expect the knowledge gained here to extend to something else, at some point I'll probably stop trying to explain it.

At some point, though, this hypothetical veers into "I wouldn't explain X because X wouldn't happen" territory. Hypothetical results don't have to be generated in the same fashion as real results, and so may have unreal explanations.

Oooh! That would be so interesting. You could do blind trials with sugar pills vs other sugar pills. You could try to make other tests which could distinguish between homeopathic sugar and non-homeopathic sugar. Then if you found some you could try to make better placebos that pass the tests. You might end up finding an ingredient in the dye in the packaging that cures your hayfever. And if not something like that, you might end up rooting out something deep!

I suspect that if you were modelling an ideal mind rather than my rubbish one, the actual priors might not shift very much, but the expected value of investigating might be enough to prompt some experimenting. Especially if the ideal mind's hayfever was bad and the homeopathic sugar was expensive.