Having an eye on someone/something is used depending on the context. For instance a parent might say to a child "I have an eye (or my eye) on you" which means theyre watching them to see if they'll do something mischievous. In a different context but still similar, I might reassure someone "I have my eye on [them]", them being a person they care for and are worried about.
And in a business context it might mean you're watching somebody to see how they progress. Having an eye on the new guy to see how much potential he has.
Hope that helped!