What makes a strong polyadenylation signal a strong signal?
Just expanding on aandreev's comment.



Polyadenylation is initiated by binding of CPSF (Cleavage and Polyadenylation specificity factor) after its binding to the AAUAAA motif.
From the introduction of this article:


  CPSF (cleavage and polyadenylation specificityfactor) binds to the
  AAUAAA hexamer via its 160-kDa subunit and possibly its 30-kDa subunit
  (Keller et al., 1991; Murthy &amp; Manley,1992; Jenny et al.,1994;
  Barabino et al.,1997). CstF (cleavage  stimulation  factor) binds to
  the U- or G/U-rich element downstream of the cleavage site, and
  stabilizes the binding of CPSF to the AAUAAA sequence of the pre-mRNA
  (reviewed in Colgan &amp; Manley,1997; Wahle &amp; Kuhn,1997).


Any alteration in the CPSF binding site will lead to reduced polyadenylation efficiency. Polyadenylation efficiency also depends on other factors such as secondary structure and presence of proximal CstF sites.