My understanding is because the Ito's integration definition keeps the martingale property.
With Brownian motion $W(t, \omega)$ defined, to define stochastic integration in a Riemann–Stieltjes style:
$$\int_0^t f(t, \omega) d W(t, \omega) = \lim_{\| \Delta_n\| \to 0 } \sum_{i=1}^{n} f(\tau_i,\omega) \left ( W(t_i, \omega) - W(t_{i-1}, \omega) \right ) $$
, the choice to made is that which point from $[t_{i-1}, t_i]$ shall $\tau_i$ pick?
If $\tau_i = (t_{i-1}+t_i) /2 $, this is the Stratonovich integration.
But look at the $f(t,\omega)$, if it's determined, or denoted as $f(t)$, we have $\int_0^t f(t) d W(t, \omega)$ is a martingale. This is natural (by intuition) as $W(t, \omega)$ is a martingale, as a special case.
So we hope the intuition goes on, that for a general $f(t,\omega)$, the integration is also a martingale.
This lead to $\tau_i = t_{i-1}$, the Ito's integration.
On @amlrg 's comment, about choosing martingale or non-arbitrage while defining stochastic calculus, it's a bit long so I'm appending the answer here.
Well, I'm not a math guy so below is just my guessing.
My guess is that when Ito etc were building up the theory, comparing to "non-arbitrage", "martingale" might be more interesting as it has more applications, a simpler concept, and more potential to extend.
"Non-arbitrage" is about pricing in quant finance, besides that there were multiple kinds of problems related to stochastic calculus, for reference, pls consult examples given in Oksendal's book "Stochastic Differential Equations" chapter 1.
Also, martingale is simpler. To define non-arbitrage other concepts such as self-financing, conditional expectation are needed.
Martingale is also a more fundamental concept. This makes it easier to extend the Ito calculus. If the $f(t,\omega)$ does not have bounded squared variation, so long as the exceptions has $0$ Lebesgue measure, the Ito integral still works: it's no longer a martingale, but still has "local martingale" property. It's said that it can be further extend to Malliavin calculus, applying to the calculus of variations, defined on Hilbert space etc.. but that is already beyond my poor limited math knowledge.