The Cotter River and dam in the Brindabella Ranges is named after Garrett Cotter. Garrett and Michael Cotter may have been brothers, but I haven't seen any definite proof of a sibling relationship between them. However, I tend to think that it's possible as both came to Australia from the Fermoy region in Co. Cork, both were sons of a James Cotter, and various lines of Michael's descendants claim a relationship so, perhaps, the claim came from Michael himself?
There are however some things which don't support a sibling relationship. When Michael came to Australia, he was sent to "Tirranna" at Goulburn, and Garrett wasn't far away at Collector to the south of Goulburn, yet there doesn't appear to have been any contact between the two.
For example, in all the Christenings, Weddings, etc. of Michael and Garrett's children, none of the other "brother's" children acted as God Parents, Best Man, Bridesmaids, etc. which I would have expected amongst first cousins. Also, none of Michael's children were named Garrett, which would be expected if they were brothers, though Garrett did name his second son Michael (James was his first born).
So it remains possible that Garrett and Michael were brothers, but I haven't found any definite or conclusive proof of that relationship.
I think you might be a bit harsh on your Mum, the Cotter of Co. Cork were the descendants of the Sons of Ottar who were the kings of Dublin and who also controlled the Isle of Man (where they are still known as the MacCotters), and possibly also controlled Anglesey in Wales. Basically, they controlled the south Irish Sea from Dublin, Man and Anglesey till Brian Boru defeated them at Dublin, and the Welsh with a Scottish army drove them out of Anglesey.
They were also influential in the Scottish Hebrides and are associated with the clans McLeod and McLean.
The Ottars were Vikings from Norway, and they are the descendants of the Scylfings, the ancient royal family in Sweden (hence Ottar, from Ottar Vendelcrow). The Scylfings were based in Sweden but were driven out to Norway by the Goths (Geats) around 500 to 600 AD.
The "Sons of Ottar" Y-DNA project has proven the Norse origins of the Irish Cotters (see http://www.familytreedna.com/public/sonsofottar/default.aspx
). Also of interest is the Wikipedia history of the Irish Cotters (see http://en.wikipedia.org/wiki/Cotter_family
), and also Miriam Ellis's speculations on an Ottar, Uther, Arthur connection is interesting (see http://www.miriamellis.com/geneaology1.shtml
). So, go and say sorry to Mum.