What would an all-flying aileron look like, and would it be a good idea?
The word aileron specifically refers to a part of the airplane, usually, but not always, connected to the back edge of the wing, that is used to control roll and to turn the aircraft when used in conjunction with the rudder.

The British Army No. 1 was based on the Wright Flyer design, and while some early airplanes did incorporate wing-warping to bend the wing itself instead of using ailerons, the No. 1 actually incorporated separate ailerons, mounted above and in front of the leading edge of the lower wing.

Perhaps, instead of aileron, you meant to say wing or airfoil. In that case, several examples exist of modern flying-wing aircraft, such as this B-2 Spirit, which designs the entire aircraft as one large airfoil.

UPDATE
Wing-tip ailerons were utilized in some designs, such as this Stout Skycar.

NASA did testing on this aileron configuration to determine whether or not this design would counteract the adverse yaw effect of traditional aileron placement. When ailerons are behind a wing with a large upper camber, it can partially shield the aileron that is deflected up, reducing its effectiveness. Moving the ailerons to the wingtips proved to significantly mitigate this issue, although not entirely - this due to the interference between the outer edge of the wing and the inner edge of the tip aileron. Nevertheless, there were also significant problems. The NASA study showed a substantially reduced rate of climb compared to traditional trailing-edge aileron. Additionally, there was a near doubling of drag from each wing, largely due to vortex generation caused by the necessary gap between tip aileron and wing. Further, while a traditional aileron may utilize multiple hinges, a tip aileron was pivoted on a shaft, where torque loading and flex became an issue. Finally, when connected via direct linkage, researchers observed that control loads felt at the yoke were often up to 3 times heavier than for conventional ailerons.
Here is an article by Bob Whittier explaining more.