Assuming the noise is stationary (meaning its probability distribution doesn't vary over time), it can be characterized entirely by its power spectrum, recorded over a time window that's long enough to represent the lowest frequencies of interest. The amplitude-versus-frequency response curve will exhibit various slopes at different frequencies, and it's these slopes that define the noise's "color" properties.

Once the spectrum is known, synthesizing the same sound is really just a matter of starting with white noise and shaping it with various lowpass, highpass, and/or bandpass filters whose skirts match the original slopes. It sounds like the 909 took that approach, generating noise by sending the output of a maximal-length shift register through some analog filters.

If the noise isn't stationary -- for instance, if its frequency content varies noticeably as the envelope progresses through its various phases, as the instrument warms up, or simply as it ages -- then it's no longer so easy to model and reproduce the sound. That's the sort of thing that can easily happen when the design relies on defective transistors.

Agreed. The article implied the "special" transistor was used to create something like white noise. It'd be interesting to know what that noise sounded like compared to "normal" white noise.

Here's a video that explains how to go from a white noise source to something that sounds like the TR-808 snare. https://www.youtube.com/watch?v=8LWQqDHhpKw