It means "a not totally sincere hope," as in a) I was half-hoping it would rain today. This means: part of me hoped it would rain today, but part of me hoped it would not rain today. Or part of me wanted it to rain, and another part of me did not want it to rain. b) I was half-hoping the girl would say something. This means: part of me hoped the girl would say something (because I would like to speak to her), but part of me hoped the girl would not say something (because I would be too nervous to speak to her).

We usually use 'half-hoping' when we wanted something, but were not quite sure about it. It usually happens when something that could happen would have both positive and negative results for us. In the case above, the speaker knew there were bad things that might happen if 'she', referred to in the sentence, would speak, but there were also some things positive a about it. "I was half-hoping I'd get sick before the exam, so I would have an excuse to retake it."