I also recommend Ogre. Ogre can do this, it provides everything needed in regards of mesh and animation support, but not as a drop-in solution. You have to write lots of code for this to be done.
For our project we implemented something like you do. The main character and any other character can be dressed with different weapons and armor and the visuals of the character avatar change accordingly.
As a start hint for how to go about this: In your modeling tool (Blender, Maya, 3ds max, etc.) you model your avatar and all its clothes you need and rig them to the same skeleton. Then export everything individually to Ogre's mesh format.
At runtime you can then attach the clothing meshes the user chooses to the skeleton instance so that they together form the avatar. This is not hard to do via Ogre-API, but for even easier access to this you can use MeshMagick Ogre extension's meshmerge tool. It has been developed for exactly this purpose.
If you want to change other characteristics like facial features, this is possible too, as Ogre supports vertex pose animations out of the box, so you can prepare pathes for certain characteristics of the face and let the user change the face by sliders or somthing like this. (e.g like in Oblivion)
One thing to be aware regarding Ogre: It is a 3d graphics engine, not a game engine. So you can draw stuff to the screen with it and animate and light and in any way change the visuals, but it doesn't do input or physics or sound. For this you have to use other libs and integrate them. Several pre-bundled game engines based on Ogre are available though.