There's a few articles explaining why a grandmother is specially close to her son's daughter. The granddaughter has 1 pair of sex chromosomes (so 1/12 of the total) and of that pair, one of those (1/24 of the total) is an exact copy of her grandmother's. An X chromosome has been passed down intact through the two generations.
It's calculated in total the granddaughter has 26% of her DNA from her paternal grandmother, not much extra, but a little extra closeness built in. A quick look seems to suggest that the little girls only get 24% of their DNA from their father's father, but I've not verified the numbers. However they often get his surname, so hopefully that balances it all out a bit.
image from: http://genetics.thetech.org/ask/ask435
Nice animation on unrelated genetic topic about flies: http://www.dnaftb.org/11/animation.html