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I would begin by looking into fractals in general and learning about the different types of shapes that they typically have.  Then, you might begin developing a sense of what would work better.  In particular, you probably want to learn about L-system fractals.  Mascots and logos tend to be the opposite of fractal--relatively simple shapes so that they can be resized and recognizeable at any scale.  I'm sure you can re-create your logo within a fractal program, but that's not the same as re-creating the logo as an assemblage of fractals.  Good luck and have fun!

Apophysis Flames would work well for that project maybe, lots of Birdlike Flames appear at random

You might consider using a biomorph algorithm proposed by Dawkins. It essentially uses a evolutionary process to search large configuration spaces to produce desired shapes. For example the following generalization of the Mandelbrot.



The varibles are free parameters. At each selection step you produce 8 or so new sets of parameters which are slightly varied from the original, then  render all them all along with the original. Pick the one that you like best and re-iterate.

The second order Mandelbrot is probably not varied enough to produce the desired image, but you can alway go to higher powers. Throw in some trig functions etc.

As obvious as that is...I probably wouldn't have realized it, thanks!

Thank you everyone for your ideas and suggestions. I will update you and let you know how it's coming!