How Long Will Earth's Year be When Our Sun Goes Red?
The year length depends on the distance between the planet's centre &amp; the Sun's centre, not the Sun's surface. So if the Sun merely expands, Earth at 1 AU will still take a standard year to perform 1 orbit.

However, when the Sun becomes a red giant, it won't just expand. As Wikipedia mentions, red giants shed a considerable amount of mass in the form of gas and dust. It's estimated that the Sun will lose around a third of its mass over the first billion years of its red giant phase, and eventually have a mass around half of its current mass; see here for details.

It's still not clear exactly what will happen to the Earth while all this is going on. It may become engulfed, like Mercury &amp; Venus certainly will be. But even if it's not engulfed, interacting with all that gas &amp; dust is likely to affect its orbit. 

If we ignore that complication, we can easily calculate the period of a 1 AU orbit around a Sun of reduced mass. The Newtonian form of Kepler's 3rd law says that 
$$T^2= \left(\frac{4\pi^2}{GM}\right)r^3$$
where $T$ is the period, $G$ is the universal gravitational constant, $M$ is the Sun's mass, and $r$ is the mean orbital radius.

If we keep $r$ at 1 AU but halve the Sun's mass, then the period is multiplied by $\sqrt{2}$, giving us a year length around 516 days. That's using the current day length, by then the rotational period of Earth will be considerably longer.

I seriously doubt that the Earth will still be at 1 AU from the Sun, though, so please don't take that figure of 516 days too seriously!