This should be what you need:
- Using double quotes to enclose the sedcommand,
- so that you are free to use single quotes in it;
- escape backticks to avoid the expansion.
sed "s/user='mysql'/user=\`whoami\`/g" yourfile
I've intentionally omitted the -i option for the simple reason that it is not part of the issue.
To clarify the relation between single quotes and escaping, compare the following two commands
- echo 'I didn\'t know'
- echo 'I didn'\''t know'
The former will wait for further input as there's an open ', whereas the latter will work fine, as you are concatenating a single quoted string ('I didn'), an escaped single quote (\'), and another single quoted string ('t know').