Why should the frequency of heads in a coin toss converge to anything at all? Suppose we have any kind of coin. Why should the relative frequency of getting a heads converge to any value at all?
One answer is that this is simply what we empirically observe this to be the case, and I think this is a valid answer.
However, my question is, is there some simple set of principles/axioms that apply to coin tossing from which we can derive this fact?

I want to second the answer by jacobe, but also add a bit of detail.
Assume that there is some probability $p$ of a coin toss being a head. Assign the outcome head to a score of $1$ and tails to a score of $0$. Note that the average score from a large number of coin tosses is also the average frequency of getting heads.
The "expected" score from any given coin toss is
\begin{equation}
p\cdot 1+(1-p)\cdot 0
\end{equation}
Of course, this simplifies to
\begin{equation}
p
\end{equation}
do to our scoring system. The law of large numbers tells
us that, as we flip the coin more times, we should eventually
see the average approaching to this expectation value (assuming
all coin tosses are independent of each other).
Now, if your question is: why do we even believe there is a probability
$p$ of getting heads, then I'm not sure there is a justification. At least, I haven't heard it. This comes back to what jacobe said about probability vs.
statistics. The existence of $p$ and independence of events are assumptions for our model. If the statistical results seem consistent, that's a good sign for the model.
