I generally dislike 'best of' posts, since it's rarely the case that the author has properly analyzed every alternative, or that a 'best' option can be determined objectively.
It's hard to argue that this is an exception. I haven't even tried any of the Linux mind-mapping alternatives; I have no desire to now. I just ask you to try this application. Try it. You'll see what I mean. It is just... superb. It is XMind.
XMind is clearly heavily based on the Eclipse codebase (I will forever be grateful to the Eclipse Foundation for the supremely liberal Eclipse license). As such, the GUI is highly customisable to a degree that would satisfy even the most hardcore KDE advocate.
In true Eclipse style, there is an outline pane, a properties pane, and the usual comprehensive key mapping configuration.
The maps produced with the default settings are indisputably aesthetically pleasing. Unfortunately, the "markers" (clipart graphics for use in maps) included with the software aren't quite to my taste, but are perfectly adequate nonetheless.
The real benefit comes when you really start using the software properly. You'll find that everything 'just works' in an intuitive manner. Dragging a subtopic away from a central topic 'breaks' the link and converts it into a floating topic. Add graphics to anything; attach files to anything. Need to break a subtopic out into more detail? Just hyperlink it to another sheet (or separate mindmap file, or anything else).
Provide easy navigation between associated maps with hyperlinked 'floating topics'. Add custom graphics to the marker list (native is 32px and PNG is supported. You can find some to import in the Open Icon Library). Import/export options are good, including support for other mind mapping software formats; so you can work with the output of popular Android applications, for example.
The base application is FOSS; dual-licensed with EPL and LGPL. However, a few of the lesser-used and more advanced pieces of functionality are both highly expensive and proprietary. This mightn't be enough to satisfy staunch FSF supporters, but I see it as a perfectly reasonable trade-off that has resulted in a very high quality application. I'd be interested to hear your thoughts.
For my personal mind-mapping requirements on the desktop, the free version of XMind is what I'll be using for the foreseeable future.
In case this post is irreparably tarnishing my reputation in the FOSS world, let me reassure you; for my wider diagramming needs, I do, of course, exclusively use Dia!