Ubuntu – Gnome terminal is not running `source ~/.bashrc` when opened

To get bash to read and execute ~/.bashrc on startup, start it as an interactive shell:

gnome-terminal --tab -- bash -ic "set-title hey; exec bash"

Now why didn’t your approach of sourcing the file in a non-interactive shell work? I strongly assume your ~/.bashrc begins with something like this:

# If not running interactively, don't do anything
case $- in
    *i*) ;;
      *) return;;
esac

This lets it return without doing anything if the shell where it is sourced doesn’t have “i” anywhere in its $-, i.e. it’s an interactive shell. $- is a special parameter, man bash says:

- Expands to the current option flags as specified upon invocation, by the set builtin command, or those set by the shell itself (such as the -i option).