Why is there a controversy on whether mass increases with speed?
There is no controversy or ambiguity. It is possible to define mass in two different ways, but: (1) the choice of definition doesn't change anything about predictions of the results of experiment, and (2) the definition has been standardized for about 50 years. All relativists today use invariant mass. If you encounter a treatment of relativity that discusses variation in mass with velocity, then it's not wrong in the sense of making wrong predictions, but it's 50 years out of date.
As an example, the momentum of a massive particle is given according to the invariant mass definition as
$$ p=m\gamma v,$$
where $m$ is a fixed property of the particle not depending on velocity. In a book from the Roosevelt administration, you might find, for one-dimensional motion,
$$ p=mv,$$
where $m=\gamma m_0$, and $m_0$ is the invariant quantity that we today refer to just as mass. Both equations give the same result for the momentum.
Although the definition of "mass" as invariant mass has been universal among professional relativists for many decades, the modern usage was very slow to filter its way into the survey textbooks used by high school and freshman physics courses. These books are written by people who aren't specialists in every field they write about, so often when the authors write about a topic outside their area of expertise, they parrot whatever treatment they learned when they were students. A survey [Oas 2005] finds that from about 1970 to 2005, most "introductory and modern physics textbooks" went from using relativistic mass to using invariant mass (fig. 2). Relativistic mass is still extremely common in popularizations, however (fig. 4). Some further discussion of the history is given in [Okun 1989].
Oas doesn't specifically address the question of whether relativistic mass is commonly used anymore by texts meant for an upper-division undergraduate course in special relativity. I got interested enough in this question to try to figure out the answer. Digging around on various universities' web sites, I found that quite a few schools are still using old books. MIT is still using French (1968), and some other schools are also still using 20th-century books like Rindler or Taylor and Wheeler. Some 21st-century books that people seem to be talking about are Helliwell, Woodhouse, Hartle, Steane, and Tsamparlis. Of these, Steane, Tsamparlis, and Helliwell come out strongly against relativistic mass. (Tsamparlis appropriates the term "relativistic mass" to mean the invariant mass, and advocates abandoning the "misleading" term "rest mass.") Woodhouse sits on the fence, using the terms "rest mass" and "inertial mass" for the invariant and frame-dependent quantities, but never defining "mass." I haven't found out yet what Hartle does. But anyway from this unscientific sample, it looks like invariant mass has almost completely taken over in books written at this level.
Oas, "On the Abuse and Use of Relativistic Mass," 2005, http://arxiv.org/abs/physics/0504110
Okun, "The concept of mass," 1989, http://www.itep.ru/science/doctors/okun/publishing_eng/em_3.pdf
As in Ben Crowell's Answer, the concept of "Relativistic Mass" is not wrong, but it is awkward. There are several things a loose usage of the word "mass" could imply, all different and thus it becomes a strong convention to talk about the meaning of the word "mass" that is Lorentz invariant - namely the rest mass, which is the square Minkowski "norm" of the momentum 4-vector. Given its invariance, you don't have to specify too much to specify it fully, and so it's the least likely one to beget confusion.
Here's a glimpse of the confusion that might arise from the usage of the word mass. To most physicists when they learn this stuff, the first time they see "mass" they think of the constant in Newton's second law. So, what's wrong with broadening this definition? Can't we define define mass as the constant linking an acceleration with a force? You can, but it depends on the angle between the force and the velocity! The body's "inertia" is higher if you try to shove it along the direction of its motion than when you try to introduce a transverse acceleration. Along the body's motion, the relevant constant is $f_z=\gamma^3\,m_0\,a_z$, where $m_0$ is the rest mass, $f_z$ the component of the force along the body's motion and $a_z$ the acceleration begotten by this force. At right angles to the motion, however, the "inertia" becomes $\gamma\,m_0$ (the term called relativistic mass in older literature), i.e. we have $f_x=\gamma\,m_0\,a_x$ and $f_y=\gamma\,m_0\,a_y$. In the very early days people spoke of "transverse mass" $\gamma\,m_0$ and "longitudinal mass" $\gamma^3\,m_0$. Next, we could define it as the constant relating momentum and velocity. As in Ben's answer, we'd get $\gamma\,m_0$. We can calculate $\vec{f}=\mathrm{d}_t\,(\gamma\,m_0\,v)$ correctly, but not $\vec{f}=\gamma\,m_0\,\vec{a}$, it fails not only because $\gamma$ is variable but also because the "inertia" depends on the direction between the force and velocity.
So, in summary, "inertia" (resistance to change of motion state by forces) indeed changes with relative speed. You can describe this phenomenon with relativistic mass, but it is awkward, complicated particularly by the fact that the "inertia" depends on the angle between the force and motion. It is much less messy to describe dynamical phenomena Lorentz covariantlt, i.e. through relating four-forces and four-momentums and one uses the Lorentz invariant rest mass to see these calculations through.
There's no controversy about whether mass increases or not, there's controversy about what you call mass. One possible definition is that you consider some object's rest frame, and call the $\tfrac{F}{a}$ you measure there (for small accelerations) the mass. This notion of mass can't change with speed because, by definition, it's always measured in a frame where the speed is zero.
There's nothing wrong about this way of thinking, it's basically a question of mathematical axiom. Only, it's not really useful to require the rest frame, because we're constantly dealing with moving objects1. Therefore, the (I believe) more mainstream opinion is that that quantity should only be called rest mass $m_0$. The actual ("dynamic") mass is defined by what we can directly measure on moving objects, and, again simply going by Newtons law, if you e.g. observe an electron moving with an electric field at $0.8\:\mathrm{c}$, you'll notice it is accelerated not with $a = \tfrac{F}{m_0}$ but significantly slower, namely as fast as a nonrelativistic electron with mass $m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$ would. It is therefore reasonably to say this is the actual mass of the electron, as seen from laboratory frame.
1 Indeed, you can argue it's never possible to really enter the rest frame. In macroscopic objects you'll have thermal motion you can't track, and yet more fundamentally there's always quantum fluctuations.
Edit as noted in the comments, amongst physicists there will of course not really be controversy about what mass definition is meant: they'll properly specify theirs, usually just following the convention of invariant mass. That can easily be calculated for any given system, from the total energy and momentum rather than the actual movements of components (which, again, you can't track). That still leaves scope for confusion to the unacquainted though, because whether the invariant mass increases or not when accelerating an object depends on whether you consider the mass of some bigger system, say with some much heavier stationary target, or the accelerated object on its own. This may seem counterintuitive, so when hearing accounts of the same experiment based on either of these "system" definitions you think there's a controversy, when really the accounts are just talking about different things.