You can't do it visually, in a GUI-mouse fashion. The terminal is line-oriented and selection only works within that framework.
See gnome-terminal usage guide
"To select a character at a time, click on the first character that you want to select and drag the mouse to the last character that you want to select.
To select a word at a time, double-click on the first word that you want to select and drag the mouse to the last word that you want to select. Symbols are selected individually.
To select a line at a time, triple-click on the first line that you want to select and drag the mouse to the last line that you want to select."
The 'dragging' mentioned here refers to adding to the selection area, not dragging the selected text. Once you finish dragging and lift up on the mouse button, the selected text is automatically copied to the clipboard. Then you can use the middle-mouse button to paste it into your destination application.
Or, once the text is selected, use Control-Shift-C to copy, or right-click in gnome-terminal and select Copy, and then Control-Shift-V to paste, or select Paste from the context menu in gnome-terminal. These are extra keystrokes or mouse clicks. The automatic placement of the selected text onto the clipboard makes it very nice to just go paste with one mouse click. That's only one more mouse-click than the feature you're seeking.
If you want, you can add a request for this feature to the gnome-terminal package at Launchpad. I don't see any bugs there about this.
(Need link to procedure to adding feature requests/bug reports here.)
Alternatively, you can try a different terminal emulator:
List of good emulators from TechDriveIn
"List of terminal emulators you may never have heard of"
Argument - er, Discussion - on the best terminal emulator
I won't list them all here, reading the information about each from these links or your own search should help you decide which is the best one for you.
Bind following shortcut:
bind '"\C-p": "\C-e\C-u xsel <<"EOF"\n\C-y\nEOF\n\C-y"'
Now after using Crtl+P your line will be copied into clipboard. You can paste it in terminal using:
xsel
And into any X application using middle mouse button or Shift+Insert.
Best Answer
screen(1) allows you to copy/paste using the keyboard, but it requires that you run your shell/process inside
screen.To copy, hit Ctrl+A [, move the cursor (using the arrow keys) to the start point and hit ⏎ (RETURN), move the cursor to the end point and hit ⏎. The marked text is now copied.
To paste, hit Ctrl+A ]
This solution also applies to Byobu, since Byobu is just a wrapper around screen.