Newton's method is very robust once you are near a root - exactly what function you chose to run it on is not of that much consequence; it will converge quickly no matter what. You might as well just use whatever is easiest.
To be more precise, let $f$ be the function we wish to find a root of and define $g(x)=x-\frac{f(x)}{f'(x)}$ to be the function performing a single step of Newton's method. First: it should be clear that things go badly if $f'$ and $f$ share a root, so let's assume they don't - and we'll assume that $f$ is smooth near the root, for convenience.
The idea is that $g$ pulls everything near a root to the root - and to quantify how fast, we can take a look at its Taylor series at a root. Let $x_0$ be a root of $f$. If we differentiate $g$ then simplify at $x_0$ by noting that $f(x_0)=0$, we can see the following:
$$g(x_0)=x_0$$
$$g'(x_0)=0$$
$$g''(x_0)=\frac{f''(x_0)}{f'(x_0)}$$
In particular, this tells us that, once we are close enough to $x_0$, if the current amount of error is $e$, one more application of $g$ makes the error something more like $g''(x_0)e^2$ - this is pretty fantastic irrespective of what $g''(x_0)$ is - if you take a small number and square it over and over and over, it gets really small really fast - which is what we want to happen to the error. I suppose if you were really eager, you could try to make $g''(x_0)$ small by choosing $f$ carefully - but realistically, it does not matter, since it tends to be the case that running an additional step of the method completely dwarfs any gains from fine-tuning $f$.
This said, Newton's method can behave pretty badly on a global scale - even for polynomials it can do really nasty things*. Choosing your function carefully isn't likely to help you much here either, since when functions have really small derivatives, the method is unstable, but when they have rapidly growing derivatives (like near an asymptote), the method converges very very slowly - basically, this isn't a good method to look for a root without having any idea of where one might be, and fine tuning $f$ won't help you there either.
(*Example: Look up "Newton's method basins of attraction" on Google - you'll get these amazing plots of where the method converges if you run it on polynomials as simple as $x^3-1$ starting at various points in the complex plane. Basically, when it's not near a root, anything can happen)