I found only one book so far which does it right,that is "Geometry: A Metric Approach with Models" by Millman and Parker.They spend about 40 pages mostly for the definition of area and proof of the existence/uniqueness.(All the answers so far describe the same idea.)
In addition to the proof in Millman--Parker book, I know the definition with measuring grid it is better, but still boring. (I do not know a reference, but I red it in some book.)
P.S.Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts in a not boring way? Say in such a way that not all the students will sleep on the lecture?
Comments.
The real problem is to prove existence, the uniqueness is easy.
Using integral does not seem to be a good idea.
There is an approach where you write the formula for area and then proving its properties. I do not like it since it moves you to discrete geometry which is completely irrelevant and the ideas used nearly useless anywhere else (so no reason to learn this stuff). [See for example "Geometry: A Metric Approach with Models" by Millman and Parker.]
The method with measuring grid (cutting everything into small squares and counting) looks much better. One can consider this method as an introduction to integral. There is only one technical statement which has to be proved --- if you rotate square then its area does not change. The only problem is that it is not generalizable to say absolute plane or sphere...
One may define the area as a limit of $\varepsilon^2\cdot N_\varepsilon$, where $N_\varepsilon$ denotes the maximal number of points in the figure one distance $>\varepsilon$ from each other. The only hard part is to prove existence of the limit $\varepsilon^2\cdot N_\varepsilon$ for say polygons. (You can exchange the limit to ultralimit --- this way everything works smoothly, but I do not want to sale my soul just to get a def of area...)
