I'm sure there's an interesting historical answer on the subject, but I think there is an AskScience answer, too, so if it’s okay I’d like to share it.

Morning sickness isn't terribly well understood, but you're right that it can happen any time. In fact, when they ask pregnant women to keep diaries of their nausea, morning doesn't stand out at all.[1] But it is probably useful to look at identifiable triggers for nausea during pregnancy. Aversions to certain smells and tastes is a well known one. Since women who don't experience nausea during pregnancy have more miscarriages than women who do, it is generally accepted as an evolutionary adaptation to protect the fetus from potential toxins and pathogens in food.[2] (Not that I needed any help knowing I shouldn't eat milk thistle, thank you very much, and now my garden is overrun.) Obviously, we encounter such things throughout the day, so it’s no surprise that pregnant women might experience nausea throughout the day, too.

But food aversion isn’t the only observable trigger. Nausea during the first trimester is also more common after fasting, in which case it improves with eating.[3] Most people fast at least once a day: while asleep. I suspect what we have here is something of a synecdoche, where a term for part of something is used to evoke the whole. We have an occurrence of nausea during pregnancy that is easy to label, amid a whole complex of nausea and vomiting during pregnancy that’s hard to fully explain to this day.

1. Lacroix, Eason, and Melzack (April 2000). “Nausea and vomiting during pregnancy: a prospective study of its frequency, intensity, and patterns of change”. American Journal of Obstetrics and Gynecology. 182 (4).

2. Flaxman and Sherman (2000). “Morning sickness: a mechanism for protecting mother and embryo”. Quarterly Review of Biology. 75 (2).

3. Jednak et al. (October 1999). “Protein meals reduce nausea and gastric slow wave dysrhythmic activity in first trimester pregnancy”. American Journal of Physiology. 227 (4).