Usually a back-reference is either $1 or \1 (backslash one) for the first capture group (the first match of a pattern in parentheses), and indeed Sublime supports both syntaxes. So try:
my name used to be \1
or
my name used to be $1
Also note that your original capture pattern:
my name is (\w)+
is incorrect and will only capture the final letter of the name rather than the whole name. You should use the following pattern to capture all of the letters of the name:
my name is (\w+)