What does the variable $\tau$ mean in frequency modulation?
It's just a dummy integration variable. The time variable $t$ is the upper integration limit, so you can't use it as the integration variable. An expression like
$\int_0^tm(t)dt\qquad(???)\tag{1}$
doesn't make sense, whereas
$$\int_0^tm(x)dx\tag{2}$$
does.
Take a look at this question and its answers over at math.SE.