The bits per sample will affect the dynamic range of your receiver.
There's a lot of math that I'm sure you can find, but here's the intuitive explanation:
A digital signal can represent only discrete quantities, where an analog signal can represent infinitely many quantities between any two discrete quantities the digital signal might represent. The difference between the actual analog signal, and the represented digital signal, is an error, and is called quantization noise.
If you have more bits, the error, and the noise is less. Of course, if there are larger sources of noise (the preamplifier, ambient RF noise, etc) then the additional noise introduced by quantization doesn't matter.
Thus, it makes sense on as SDR to adjust the preamplifier's gain so that the noise floor is just above the quantization noise. Otherwise, you are wasting a lot of bits just carrying noise.
This maximizes the bits available for actual signal, which is good, because it also maximizes dynamic range, or how much power can be received by the receiver before it clips (the digital signal can represent a signal only so big). This is of particular concern for SDRs because of their wide receive bandwidth: you may not be listening to that guy with the S9+ signal, but your receiver is. If that signal is strong enough to hit the limits of your ADC, then you get clipping, which will introduce harmonics all over the band, making you unhappy. You could turn down the receiver's gain, but then the very weak DX station that is just above the RF noise floor will be below the quantization noise.
The resolution (number of bits) for the DAC is not really as critical, since you don't need a lot of dynamic range to transmit. You will still have quantization noise, but since the digital signal can trivially be scaled to use the full range of the DAC, the noise is usually negligible. An approximate model is the noise floor will be 6 dB below the signal for each DAC bit. For an 8 bit DAC this gives a noise floor of -48 dBc, sufficient for many applications. Further improvements can be made by noise shaping combined with analog filtering of the output. Or simply upgrading the precision of the DAC may be the most economical solution: DACs are cheaper than ADCs.
So, 8 bits is more than sufficient for experimentation, but this is definitely a "more is better" situation. It all depends on the performance you require, and also your ability to build the rest of the receiver to utilize the full capabilities of the ADC.
As mentioned by others in comments and answers, it is also possible to reduce the quantization noise by oversampling the signal of interest, then decimating the resulting data. For example, say we are interested in signals up to 30 MHz, so we require a 60 MHz ADC. If we use a 240 MHz ADC, but the analog input has no frequencies above 30 MHz, then we have four times more samples than required.
We could simply keep only 1 in 4 samples, but if we first use a digital low-pass filter we can actually remove some of the quantization noise. This is because the quantization noise will be distributed through the spectrum: from 0 to 120 MHz, the maximum at our 240 MHz sample rate. Low-pass filtering then removes the noise between 30 MHz (our upper range of interest) and 240 MHz, so we've effectively removed some of the noise by averaging it out.
The math works out so that for every 4x in excess samples, it effectively adds another bit to the resolution, or increases dynamic range by 3dB. For example, an 8-bit, 240 MHz ADC is equivalent to a 9-bit, 30 MHz ADC.