How to update a popup content from its marker?

_popup has an underscore in front of it to indicate that it's a private / member instance, and should not be accessed directly. The correct API is Layer.setPopupContent(). e.g. marker.setPopupContent(newContent);

I suppose you want the content to be changed after some event occured, like mouseover, contextmenu, or anything else.

To do so, you can use the following code:

//marker creation
var marker = L.marker([44.63, 22.65]).bindPopup('something').addTo(map);
marker.openPopup();

//changing the content on mouseover
marker.on('mouseover', function(){
    marker._popup.setContent('something else')
});

As you can see, you can acccess the popup for the desired marker using marker._popup method, and then use setContent method to change the text inside it.

popup.setContent method reference

Here's some code on Plunker demonstrating this: http://plnkr.co/edit/vjS495QPXiJpKalrNpvo?p=preview

Might be to late to reply, but for others, I think the best ways is here

http://jsfiddle.net/cPTQF/

$('button').click(function() {
   // Update the contents of the popup
   $(popup._contentNode).html('The new content is much longer so the popup should update how it looks.');

   // Calling _updateLayout to the popup resizes to the new content
   popup._updateLayout();

   // Calling _updatePosition so the popup is centered.
   popup._updatePosition();
   return false;
});