[...] then started reading on different forums that increasing the length of your controller's cable can introduce a small amount of input lag.
While it's technically true that any added transmission distance adds lag, citing it for this case is bullshit of the know-it-all kind.
Can this lag (in milliseconds) be modeled as a function of cable length (in meters)?
Yes, except, with distances in meters the time unit will be nano seconds (ns). That's millionth of microsecond. So looking at the dimensions involved should already give a good hint how insignificant this is - at least for controllers.
If I have a cable that is x meters long, can I predict with reasonable suspicion that the input lag will be y milliseconds? If so, what is this equation/relationship?
Signal speed is defined by the Velocity Factor of it's transport medium, a number between 0 and 1 to be multiplied with c, the speed of light.
Copper cables can provide factors between 0.5 and 0.95, although the later are exceptional odd constructions not really usable in most situations. It's all about the wire layout, material and insulation. An open air wire will rate better than a coax cable which in turn is better than a able used for controllers.
Average copper cabling used in electronics range around 0.7, so a signal in a copper cable will travel at ~70% of light speed or ~210,000 km/s.
For all 'practical purpose' using a factor of 2/3 is quite convenient considering that sped of light is 300,000 km/s. Using that and the power of the metric system gives the nice values of 20 cm/ns (*1) or 5 ns per meter.
Using this means a 2m cable, which should easy help to relax your arms will add 10 ns, ot 1/100,000 of a millisecond delay to your button press.
Bottom line: Yes, it's a delay, but absolute insignificant in relation to the several milliseconds your body already adds.
Now, being at this point of optimization, it may be way more interesting to see what influence the travel length of a button press has, or how much time it takes your skin to deform until it transfers enough pressure to make the button close a circuit ... a whole world of more or (usually) less useful optimizations opens :))
*1 - 8 inches per nano second or for the metrical challenged or 4.5 ns per tripple-foot, ofc, only when using certain shoes :)