The bulk composition of the Moon's crust. (This one does not actually involve the Earth, but I feel it is still important to mention.) The Moon's mantle and crust chemical composition could be explained if the Moon had a large iron core, but its core is actually quite small.
The ratio of the Earth and Moon's mass far exceeds any other planet in the solar system, and this begs the question of how did so much material become in orbit of the Earth. (Not evidence as such, but raises the question in the first place)
Getting more indirect...there are bands of warm silica-rich dust orbiting nearby stars which is interpreted as planet-sized bodies having collided with each other, so there is precedent for similar events happening elsewhere in the galaxy. (Again, I realise this is one is not strictly answering the question, but Earth Scientists often have to look well outside the Earth to answer their questions!)
Hope this gets you started!
Update: I recently came across this short article, which provides a summary of some of the latest thinking on the Moon's formation, including the Giant Impact Hypothesis.
Sources: de Pater and Lissauer (2010): Planetary Sciences; Canup and Asphaug (2001), Nature
Following what I stated above, it's easy to see where something similar would have happened had Earth been struck by a Mars sized object; only the damage would have been much more severe as I stated above.
Is it possible that the moon grazed Earth in the distant past prior to settling into a stable orbit? That's a more likely scenario than a head-on collision in my opinion. There still would have been a lot of interaction between the two bodies; including highly electrical interaction. Thus, that could easily explain our geological findings as well as possibly the tidally locked situation.