Is it permissible for me to perform Hajj on my own behalf and for my friend to perform Hajj on behalf of my deceased mother in the same year, knowing that my friend has previously performed Hajj for himself?

The majority of scholars permit performing Hajj on behalf of the deceased, provided that the proxy has already performed Hajj for themselves. Therefore, if your friend has performed Hajj for himself, there is no impediment to him performing Hajj on behalf of your mother.