The term "white noise" in economics is derivative of its meaning in mathematics and in acoustics. To understand the economic significance of white noise, it's helpful to look at its mathematical definition first.
White Noise in Mathematics
You've very probably heard white noise, either in a physics lab or, perhaps, at a sound check. It's that constant rushing noise like a waterfall. At times you may imagine you're hearing voices or pitches, but they only last an instant and in reality, you soon realize, the sound never varies.
One math encyclopedia defines white noise as "A generalized stationary stochastic process with constant spectral density." At first glance, this seems less helpful than daunting. Breaking it down into its parts, however, can be illuminating.
What is a "stationary stochastic process? Stochastic means random, so a stationary stochastic process is a process that is both random and never varying -- it's always random in the same way.
A stationary stochastic process with constant spectral density is, to consider an acoustic example, a random conglomeration of pitches -- every possible pitch, in fact -- which is always perfectly random, not favoring one pitch or pitch area over another. In more mathematical terms, we say that the nature of the random distribution of pitches in white noise is that the probability of any one pitch is no greater or less than the probability of another. Thus, we can analyze white noise statistically, but we can't say with any certainty when a given pitch may occur.
White Noise in Economics & in the Stock Market
White noise in economics means exactly the same thing. White noise is a random collection of variables that are uncorrelated. The presence or absence of any given phenomenon has no causal relationship with any other phenomenon.
The prevalence of white noise in economics is often underestimated by investors, who often ascribe meaning to events that purport to be predictive when in reality they are uncorrelated. A brief perusal of web articles on the direction of the stock market will indicate each writer's great confidence in the future direction of the market, beginning with what will happen tomorrow to long-range estimates.
In fact, many statistical studies of the stock markets have concluded that although the direction of the market may not be entirely random, its present and future directions are very weakly correlated, with, according to one famous study by future Nobel Laureate economist Eugene Fama, a correlation of less than 0.05. To use an analogy from acoustics, the distribution may not be white noise exactly, but more like a focused kind of noise called pink noise.
In other instances related to market behavior, investors have what is nearly the opposite problem: they want statistically uncorrelated investments to diversify portfolios, but such uncorrelated investments are difficult, perhaps close to impossible to find as world markets become more and more interconnected. Traditionally, brokers recommend "ideal" portfolio percentages in domestic and foreign stocks, further diversification into stocks in large economies and small economies and different market sectors, but in the late 20th and early 21st centuries, asset classes that were supposed to have highly uncorrelated results have proven to be correlated after all.