I would be concerned about measuring area by using a mower unless you are superman and can keep the exact same amount of overlap each path and figure out how to calculate for the turns.
I don't think you need to be a superman to keep up an overlap constant. It's a matter of experience, especially for those who mow much with both, either side mounted mower or rear mounted mower.
Yes, I was thinking about the turns. In case of any kind of odometer, it needs that the mileage should be measured by the rear RH side wheel i.e. the closest one to the centre of the turning radius. I'm guessing all of us are turning to the right side while mowing. And the sharper the turning angle, the lower the angular velocity of the RH side wheel. So the mileage measured by that wheel could be closed to real on the turns. I don't say it'll be exactly real. Only a few passes on the same lot will show you, how precisely will the mileage be measured. But I'm sure it'll be more precisely measured then by GPS device, especially on the hilly lots or in the mountains.
I'm contemplating on the Bike Odometer.
The following link is just for example:
Waterproof Bicycle Bike Cycle Wireless LCD Digital Computer Speedometer Odometer | eBay
Look at the price! If it doesn't fit a tractor, you can simply throw it away... or give it to your kid.
I was talking to sales person about it. To install it you need to attach a sensor to the axle casing and to stick a wheel magnet to the rotating axle so to provide a gap of 5 mm (~1/5"). And it'll start measurement. Salesman only doubted, would the system work if the speed will be comparatively low (I mean mowing speed). He said he's sure, no problem if that speed exceeds 5 km/h (~3 m/h) - it's nearly the walking speed of a human, if less - who knows.
Another one I was thinking about is a Hub Odometer (Hubometer).
Link here:
http://www.mytractorforum.com/showthread.php?t=219889
What I don't like with it is, that it could be simply broken by the shooting out stones from the side mower discs.
Still searching for the best solution and good ideas on this subject.