Why does the Lewis definition of acids specify a PAIR of electrons and not a single electron?
It partially has something to do with the history of acid-base definitions. We started with the Arrhenius definition which was based on the generation of $\ce{H+}$ or $\ce{OH-}$ in aqueous solution, and then to the Bronsted-Lowry definition which moved towards acceptance or donation of $\ce{H+}$. Finally came the Lewis Definition which removed $\ce{H+}$ and left us only with electron pairs, to be consistent with the past two definitions. We couldn't just say electrons because that would have included a large class of reactions called redox reactions which are clearly not the same as acid-base.

Now, with respect to your answer to the multiple-choice question:


  I thought the answer was $\ce{NH4+}$ as it could accept an electron, but the Lewis definition specifies accepting a pair of electrons.


There are two points that I would like to draw your attention to:


You're right in that $\ce{NH4+}$ is an acid (more specifically we call it a conjugate acid), but by the two older definitions.
However, $\ce{NH4+}$ cannot accept even one electron in this case. If you draw the Lewis structure, you'll find that although there is a net positive charge, all hydrogens have 2 electrons and the central Nitrogen has eight, fulfilling the octet rule.