Just to reiterate what everyone else said:
SN1 works with polar protic solvents; those capable of hydrogen-bonding.
As for what makes a good leaving group, it depends on the conjugate base of the acid. The conjugate base of a very strong acid (HI, HBR, HCl in that order) have very little to none in basic properties. So, to give you an idea, those would be best. Something like an amine though, RNH2, that would be a conjugate base of an extremely weak acid, and would therefore be a very good base. It can also be stabilized by resonance due to the lone pair on the nitrogen atom; this would make for a bad leaving group.
Also, just to note, the conjugate bases of weak acids are weak bases. If you had to compare a bunch of different compounds though, then generally, it's an inverse relationship; the stronger the acid, the weaker the conj. base, and vice-versa.