I'm not a hundred percent sure as prepositions are always tricky for a non native but I would see the difference as follows:
1a. An issue is what democratization means for the rural poor.
--> = what effect it has on the poor
1b. An issue is what democratization means to the rural poor.
--> = what it means to the poor themselves
2a. Democratization is important for me.
--> = important for my life (I think there are only few times where "for" is used with important: e.g. "it is important for me to eat healthy food". I'd usually go with "sth. is important to me" in the sense of 2b.)
2b. Democratization is important to me.
--> = it means a lot to me
3a. The opinions of the critics meant nothing for him.
--> = they had no effect on him
3b. The opinions of the critics meant nothing to him.
--> = he didn't care about them
4a. She meant so much for me.
--> (I don't think this is correct at all!)
4b. She meant so much to me.
--> = I loved her very dearly