Divide the exponent by 4 to find the remainder.

We need to find the remainder of $1265 \div 4$.

$$1265 \div 4 = 316 \text{ with a remainder of } 1$$

This can be written as $1265 = 4 \times 316 + 1$.

We can rewrite $i^{1265}$ using the remainder:

$$i^{1265} = i^{4 \times 316 + 1}$$

Using the exponent rule $a^{m+n} = a^m \cdot a^n$ and $(a^m)^n = a^{mn}$:

$$i^{1265} = (i^4)^{316} \cdot i^1$$

Since $i^4 = 1$:

$$i^{1265} = (1)^{316} \cdot i$$ $$i^{1265} = 1 \cdot i$$ $$i^{1265} = i$$

The calculated value is $i$, which corresponds to option A.

The final answer is $\text{A}$.