Unfortunately, trying to defeat side-channel attacks by adding random noise usually only increases the number of samples required to extract information, rather than preventing the attack entirely. (You can blame the central limit theorem for this.)
There are also other (unrelated) techniques that use randomness to eliminate side channels. One of the most basic anti-timing-attack countermeasures is to use RSA blinding in which a base C is first randomized by computing C^r mod N before that (random) result is combined with the secret key. The randomness can then be removed from the final result. This defeats attacks that depend on choosing or knowing the value C.
Eg. my app is performing digital signatures and I'm sure that they take <1ms CPU time, but performing digital signatures in 10ms is acceptable for my application, so when I perform a signature I measure the CPU time elapsed, say 0.5ms and then wait for 9.5ms.