[SamBam]

Actually the sun still edges out Mercury.

Think of it this way: If we take the Earth as stationary and just look at the respective motions of the Sun and Mercury, then the Sun is also (roughly) stationary* and Mercury moves around and around it, sometimes close to us and sometimes far.

Now, if Mercury actually yo-yo'ed through the Sun, then you'd be right: exactly half the time it would be closer to us, and half the time it would be further from us.

But it doesn't yo-yo through the Sun, it moves in a circle. When it's 90º from us and the Sun, it's still further away from us than the Sun is. So it has to get even closer before it's equidistant. So it's actually closer to us only less than half of the time.

*Yes, the Sun would also appear to orbit around the gravitational center of mass, but this doesn't affect the thinking above.