How to create a noise budget and what to do with it afterwards?
A noise budget in this context seems to be the minimum resolution you need in order to obtain the information you want.  

For the ADC example - if you want to be able to resolve 0.25v out of a 5v signal then your signal-to-noise ratio is 5v/0.25v = 20. To achieve this signal-to-noise ratio you'll need an ADC with at least 5 bits of resolution - 2^5 is 32. 4 bits will not work because 2^4 is 16 which is less than 20.  

To summarize - to pick your noise budget, figure out what is the minimum signal you need to resolve.  Then calculate the lowest bit rate you'll need in order to resolve that signal.

Of course when working with analog circuits you have to take the non-linearity of the circuits into account.  The Wikipedia entry on ADCs has a great summary of things to consider:
http://en.wikipedia.org/wiki/Analog-to-digital_converter

Bottom line: You generally should pick an ADC with slightly higher sampling than the minimum you require.

For completeness I'll also mention circuit noise.  Circuit noise is the noise generated by the components within a circuit. Examples include thermal noise, shot noise, and 1/f noise.  These place a minimum level at which you can resolve a signal and increasing the resolution of the ADC in a circuit will not help with this.  More information on the Wikipedia page:
http://en.wikipedia.org/wiki/Noise_(electronics)