The first is I.S.S. Baby. As a fan of Ice Ice Baby myself, I have to say this is my favorite. The lyrics are included in the description on the video's YouTube page. It was made by a group of NASA co-op students.
From their YouTube description:
Although they may work on trajectories, structures, hypersonic fluid flow, rocket engines, robots, or space suits during the day, the co-ops at NASA are more than just engineers. They are more than mere nerds. They are ambassadors to their generation for NASA. They can sing. They can dance. They can even rap (OK, not really).
Do the Cosmonauty will wow you with its mesmerizing dancing cosmonauts and not-so-subliminal attempt to make you want "borsch" (misspelling of borscht):
Next is Astrobiology Rap. It's all about the origin of life and search for life in the universe, but it's kind of long and monotonous.
For something more serious, check out Apollo Guys. The ballad tribute honors the engineers who made Apollo such a success. It was made by ReelNASA, the same group of NASA co-op students who made I.S.S. Baby above.