The correct answer is $\boxed{\text{A}}$.
The angular momentum of a body is given by the equation:
$$L = I\omega$$
where $I$ is the moment of inertia of the body and $\omega$ is the angular velocity of the body.
In this case, we are given that the moment of inertia of the body is $I = 1\text{ kg m}^2$ and the angular velocity of the body is $\omega = 2\text{ rev/s}$. We can convert the angular velocity to radians per second by multiplying it by $2\pi$, so $\omega = 2\pi\cdot 2\text{ rev/s} = 4\pi\text{ rad/s}$.
Substituting these values into the equation for angular momentum, we get:
$$L = I\omega = (1\text{ kg m}^2)(4\pi\text{ rad/s}) = 4\pi\text{ kg m}^2\text{ rad/s}$$
This is equal to $\boxed{1.257\text{ kg m}^2\text{ rad/s}}$ in SI units.
Each of the other options is incorrect because it does not represent the correct value of the angular momentum of the body.