Sounds to me like what you really want is some sort of "meta-schema", a database schema which is capable of describing a flexible schema for storing the actual data. Dynamic schema changes are touchy and not something you want to mess with, especially not if users are allowed to make the change.
You're not going to find a database which is more suited to this task than any other, so your best bet is just to select one based on other criteria. For example, what platform are you using to host the DB? What language is the app written in? etc
To clarify what I mean by "meta-schema":
CREATE TABLE data (
id INTEGER NOT NULL AUTO_INCREMENT,
key VARCHAR(255),
data TEXT,
PRIMARY KEY (id)
);
This is a very simple example, you would likely have something more specific to your needs (and hopefully a little easier to work with), but it does serve to illustrate my point. You should consider the database schema itself to be immutable at the application level; any structural changes should be reflected in the data (that-is, the instantiation of that schema).
