Ah. Every once in a while, I see a question come up that is interesting. In this case, it should not be difficult to do. In fact, I can see at least one solution, and maybe a second way to do so. (I'll post an answer later today. Must run out now. Sorry, but at least you can know to expect an answer if nobody else gives you one.)
A question first though. Since the mean of a variable has no impact on the correlation, do you care what the mean of y will be? Or can it simply have mean 0?
Next, I assume you mean the traditional correlation coefficient, thus the Pearson version?
Hmm, as I think, a more interesting question is, given a set of n variables, with their own set of inter-correlations, is can we choose a new variable that has a given set of n specified correlations with each of those n variables? And I think the answer is yes, of course we can do so, as long as we have sufficient degrees of freedom.
Before I go for now though, here is a fun paper on the subject.
Later... (m-file solution attached to this answer.)
The basic idea is for a variable x, find a new vector y0, such that y0 is orthogonal to x. Then choose some linear combination of x and y0 that has the desired correlation.
