What does an oscilloscope's input resistor and capacitor do?
It would be really nice if a scope input had infinite resistance and zero capacitance but that, unfortunately is impossible. Sensitive input amplifiers will always have a small amount of input capacitance and, there will always be a little leakage current from an amplifier's input. Don't forget the scope lead too - it might be a metre long and easily introduce 10 pF.

A 1 Mohm resistor might be enough to convert the leakage current to an offset of a few millivolts i.e. sufficiently small so as not to give a false measurement of any significance. So, with 1 Mohm and 1 nA leakage you get a millivolt offset change in the scope when you connect the probe tip and earth together. There is also the issue of noise - you would be unlikely to be impressed if the probe was unconnected and you saw 100 mVp-p of ripple on the display.

The 1 Mohm resistor and (say) 15 pF capacitor form a low pass circuit when the probe is unconnected and, subsequently have a noise bandwidth of about 15 kHz. Given that your scope analogue channel might have a noise of (say) 10 uV/\$\sqrt{Hz}\$, the ripple will be about 1 mV RMS or about 6 mVp-p (six sigma calculation). It's much more complex than this to analyse but hopefully, my simple calculation hints that there are other things to consider that might give the impression the scope's performance isn't that good when the probe is unconnected to a circuit.

Added to this is the need for all scopes to standardize between manufacturers means 1 Mohm is commonly accepted.