How the Gain Term $ K \left( n \right) $ Is Derived? Why Is It Called Gain?
In all adaptive signal processing schemes, be it a Least Mean Squares (LMS), Recursive Least Squares (RLS) or a Kalman Filter, The fundamental concept is the update of some parameter: such as the vector filter coefficients of a Transversal Tapped Delay Line (an FIR indeed) structure of LMS or a state of a dynamical system as in the Kalman filter.

The generic structure of the update mechanisms resembles the following:

 parameter(n+1) = parameter(n) + K(n)*errorfunction(n)


in which the thing called parameter is updated from its previous value at index n to its current value at (n+1), based on a correction term that is some meaningful function of error (denoted by errorfunction(n)) and weighted by a specific multiplier K(n) commonly referred to as the GAIN.

As you can see, the gain multiplier, determines the amount of correction that will be applied to the current value of the parameter so as to update it to its next value. That's why its called as the gain. The larger the gain K(n) the more correction there will be hence more responsive will the filter be untill it becomes unstable due to too much gain etc... The gain K(n) shall be made variable based on the convergence properties of the filter.

The specific form and derivation of an optimal gain term is completely dependent on the filter structure and the necessary mathematical manipulations of the signals inside the processing blocks, so as to get the required update equations...