Very quickly cos I'm doing oral exams all day....
Re placebo effect on children - there has been some research into placebos used on children and animals (which I have not seen) and the thinking is that the placebo effect does still work. The mechanism is that it works on the carers who either communicate their newfound calmness and expectation of recovery to the child/animal, or change their view of the symptoms so that they think they are getting better, or any combination of those two. I agree that that is harder to believe than the direct placebo effect on people who themselves expecting a cure, but I think it's just about plausible. Will have to find the research and see if it looks any good!
I suppose another possibility might be that the baby stopped crying cos you gave it something sweet.
When we were a bit older (not relevant to the baby point), my mother used to give us a vitamin c tablet for any ailment (except a wasp sting, for which you got a glace cherry). Worked wonders!
Eve I don't quite get what you mean about 'placebo effect can't cure something you weren't taking the medicine for in the first place' - why not? After all, you might have been expecting a general sense of improvement from taking the pills, which would have contributed to your headaches going away. Or the headaches might have been partly caused by anxiety about your general health, and when you took the homeopathy it removed the anxiety and hence the headaches. (Or, of course, the headaches might have gone away by themselves - that's exactly the kind of change that does happen when you're pg.) Either way, it doesn't really matter, as the important thing is your headaches went away in a perfectly safe way and you did not end up taking any medicines that might have harmed the baby. 
Anyway most people on this thread seem to be pretty much agreed - in terms of the OP, it's unlikely to do any harm and might help you cope, but it would be a good idea to consider other options.