There is a hypothesis that says a part of the Earth was split away and became the Moon. Is there any scientific evidence for this claim?
There is quite a bit. Here are the ones that come to mind immediately:
Similar surface ages - The oldest rock on the earth and the oldest rocks returned from the moon are the same age. This implies similar creation time.
Isotopic composition - The ratios of various atomic isotopes are basically the same indicating that the two bodies were created from the same original material
Relative densities - The moon is overall composed of material that is lower in density than that of the Earth. This is to be expected from the impact scenario as it is the lighter materials from the Earth's crust that would have been blasted off to form the moon.
Lack of lunar volatiles - The moon is missing a lot of the lighter, low boiling point volatiles that are expected to be trapped in its rocks. This too is to be expected from the collision scenario as they would have been vaporized.
The fission hypothesis does not really explain why the moon rotates at the same rate as it's orbit of the Earth. The scenario of a external force like another planet or giant comet hiting the Earth and braking off a piece that created the moon would better explain the velocities of rotation. As they say the second hypothesis is not ruled out, but the first just makes better sense when you study the rate of velocity of its orbit around us and the rotation on its axis.