The ז' מצות בני נח
This is really a misnomer as these were actually commanded to אדם הראשון. If so, why are they called the ז' מצות בני נח? The answer is based on the Rambam in Hilchos Melachin 9:1&2. The Rambam writes:
על שישה דברים נצטווה אדם הראשון--על עבודה זרה, ועל ברכת השם, ועל שפיכות דמים, ועל גילוי עריות, ועל הגזל, ועל הדינים.
הוסיף לנוח אבר מן החי, שנאמר אך בשר, בנפשו דמו לא תאכלו
We see from the Rambam that אדם הראשון was only commanded on 6 of the mitzvos, the 7th was only given to Noach. The reason is very simple, אדם הראשון was not allowed to eat meat so therefore the prohibition of אבר מן החי would have made no sense for him. He could not even eat meat that was not אבר מן החי (e.g. meat from a dead animal). Because of that we can't call them the ז' מצות בני אדם because אדם was only commanded on 6. Therefore we call them the ז' מצות בני נח because Noach was the first person to be chayav on all 7.
The Rambam is basing himself on the medrash at the beginning of Parshas Mishpatim.
The Ran (Sanhedrin 56b) points out that the Medrash and Rambam seem to be against the Gemara. The Gemara learns out the prohibition of אבר מן החי from the pasuk מכל עץ הגן אכל תאכל, they were allowed to eat the fruits but not אבר מן החי. From the gemara it sounds like אבר מן החי was given to אדם just like the other 6 mitzvos. The Ran explains that there is no contradiction, the prohibition of אבר מן החי is really learned out from the pasuk in Noach and was not given to אדם at all. However, because the gemara learned out the other 6 from the pesukim in Bereishis it said that אבר מן החי is also from there as well (like an asmachta) even though in truth the issur is really from the pasuk in Noach.