We should be able to work this out fairly simply. These figures are not exact, but the maths should hold true. As we're just looking at a 'slice' around the earth, we can treat this as a normal circle.

Assuming the circumference of the Earth at the equator to be 40,075,020m (figure from wikipedia - it'll do for this purpose, surely?)

And knowing that Circumference/π = diameter

we can see that 40,075,020/π = 12,756,275.1

If we then add 1 meter to our circumference

we get 40,075,021/π = 12756275.4

So if the string (and therefore circumference) is 1 meter longer, it would give a diameter 0.3m wider. If the string was hovering, this extra would be shared on either side of the earth, and therefore would be 0.15m, a mere 15 cm, off the ground.

That's the long winded way of going about it, but as Daveshorts has just pointed out to me it's easier to say that the circumference is 2π times the radius, so if you increase the circumference by 1m you will increase the radius by 1m/(2π), which just happens to be 0.16m.