For "quality of being distinct":
* -ness: distinctness (a state of being distinct)
* -ity: distinctivity (the quality of being distinctive)
For "act of distinguishing":
* -tion: distinction (the act of recognizing a difference)
For "marked by distinction":
* -ed: distinguished (having achieved eminence)
For "one who distinguishes":
* -er: distinguisher (someone who identifies differences)
So, the specific suffix you choose depends on the specific meaning you want to convey.