As I understand it (And I could be dead wrong) such additions typically take the form of something painted over the basic sprite (A sprite on a sprite). In the case of a 'weapon', you would take the base animation making the motion you want for his weapon swings, then on a separate layer you would animate the weapon's motions (Being careful to ensure that bits of said weapon disappear if they should move behind the character. Armor would be done in the same way, painted over for every type of different armor design.

Which, as you might imagine makes custom weapons and armor -very- difficult to get right if the custom armor has a different shape from the original sprite's armor.

I believe that's one of the primary driving forces behind things like FFT where only shields/weapons were given special artwork during animations. If you know where your sprite's hands will be through the whole animation, putting a weapon in them is easy.

The only commercial game I can think of that allows complete customization (And uses sprites) is Terraria, and if you want it being played, the motions are simple enough (Always animated such that the weapon will be closest to the screen and swung across the same axis) that doing it doesn't cause the problems I just mentioned.

I started making a 2D RPG once, but I never got too far. I did however manage to get equipment working. I used this site:
http://charas-project.net/charas2/index.php
to generate different parts. Each sprite then had a body, a face, clothes, hair, hats, e.t.c. Whenever a character's equipment changed I would pre-render all the layers of the sprite to a BufferedImage. Each frame I only had to draw a single image, just like usual.

Yes, just create a BufferedImage, create a Graphics2D from it and draw all the layers in order on it. Then just draw the complete sprite as you normally would. You can obviously do this with your own sprites too, not just the ones from the site I posted.

You can put a special pixel on each of the character and equipment sprites, such as a reserved alpha value, and make them line up like a socket and a jack.  Draw the pixel in the appropriate place for each animation frame, then bake it in (you don't want to hunt for these pixels at runtime).  That'll take care of different offsets, there's really no getting around the fact that you have to custom-craft the equipment and characters to go together, especially with pixel art.  You have to with 3d art too, but it can be a little more forgiving.  It's incredibly long, slow, and tedious work, and there's a reason so many games have such a high budget for art assets.