I've had a look at the PRO Victoria index of passenger arrivals, which was the usual port called at (before Sydney), and there were dozens of Edwards arriving in the time frame. I don't think there are any short cuts, especially when Australian certs are so informative and Christi's ancestor's (unknown) name will be given on her children's birth certs, at her marriage, and of course on her death cert. And depending on which state she was living in, her certs will show (some or all of) her parents' names and their address, how many years she had lived in that state, her (exact) place of birth, and many other details of her life.
If Christi gives more family names, places, and dates it might be possible to get more information from indexes.

Sue