The best way I've found to imagine a timing attack is as a set of graphs produced from thousands to millions of data points, each of which is a time measurement. Any given high-level input takes a certain amount of time to process (example: trying a login). Curves - and thus timing side channels - are distinguishable from one another when the curves have different peaks. The width of the (probably bell-shaped) curves are going to be determined by the amount of variance.
What happens if you add a little predictable random variance to each operation that's the same in each operation? Each curve probably gets a little wider, but you don't move their peaks relative to one another. At most, it might mean the person carrying out a timing attack needs to gather more data. Timing attacks already work with network noise implicitly doing what you've suggested doing explicitly.
You can simulate this yourself. Graph a thousand points of [0-1] and another thousand points of [0-1]+0.1. Superimpose the curves on one another. You may have to use some bucketing to make the curves visible.