The easiest approach if acceptable is to make the bias point adjustable. Also choosing good quality components and Mosfets that has been designed for the type of circuit you are designing i.e. Audio. You could look at auto bias circuits but that's probably over kill for what you want. If you have a circuit in mind post it here and we can have a look.
Considering this circuit, what parameters should be kept in mind while designing the Q-point for maximum stability? I remember reading somewhere that Rs stabilizes the Q-point, however I am not quite getting the logic behind it.
Actually it is just a question in one of my text book and I am having difficulty answering it.
The way to stabilize any circuit is by adding negative feedback (Google). In short: negative feedback will use a part of the output signal to counter any change in the input signal or in circuit parameters.
In your circuit, look at RS. Assume all voltages are stable (reasonable). One typical issue with MOSFETs is a comparatively large variation in threshold voltage (Vt).
Assume RG, RS and RD are fixed such that the desired operating point is achieved for a typical value of Vt. What happens if Vt is lower than the typical value? In which way will this influence the drain-source current and therefore the current through RS? In which way will the change in current through RS influence VS? And, consequently, what will this mean for Vgs and therefore in turn for Ids? Notice how a change in an output parameter (VS) closes a loop to a change in an input parameter (Vgs) which is what feedback is all about.
Perform the same route of reasoning for a rising value of Vt.