Is there a property comparable to IND-CPA for hash functions taking a random value?
Not only is your idea of using $H(x|| r)$ sufficient, it will even in general provide unconditional security, if $H$ satisfies some regularity property, which is (roughly) that the size of the preimage of $H$ with a given prefix $x$ is always approximately of the same size.

To reformulate your question, you are asking whether $H(x || r)$ leads to a hiding commitment scheme, i.e., a scheme com for which it is infeasible to distinguish com$(x_0)$ from com$(x_1)$ given $x_0$ and $x_1$.

If you use any standard hash function which maps long strings to shorter string, observe that there will be in general many preimages to any given output of the hash functions. This means that given a hash value $h$ and strings $x_0$ and $x_1$, there will almost always exist values $r_0,r_1$ such that $H(x_0|| r_0) = H(x_1||r_1) = h$ (and the number of such values should be essentially the same for $x_0$ and $x_1$). This means that it is impossible to find out whether $h$ was obtained by hashing $x_0$ or $x_1$, even if the adversary is computationally unbounded.

If you want more formal details, an improved variant of this scheme was studied in this paper (where they show that their scheme is also binding - i.e., you cannot change your mind about $x$ if you later reveal the input you've fed to the hash function - and obtain a perfectly hiding commitment scheme without even having to assume that $H$ is regular).