Select text in iTerm using keyboard

Yes! In iTerm2, do this:

From the relevant documentation

1. Cmd-F
2. Type some text that matches part of what you want to copy.
3. When you have a partial match on what you want to copy, you can expand the selection.

Press tab to grow it one word to the right; press shift-tab to grow it one word to the left.

As soon as you press tab or shift-tab, it is copied to the clipboard. You can press option-Enter to immediately paste the selection and close the find bar at once.

You could also assign custom keys in Prefs -> Keys. (Source)

Similarly settings can also be used to move line selection.

Alternatively, the autocorrect feature reduces the need for copy-paste. Just type the start of some text that is either on the screen or you have copy-pasted recently and press cmd-;. A popup window appears and you can choose from there. Autocorrect works best for a single word, though.

(I updated this answer to reflect new features in iTerm 3.0. Thanks to @sfarzy for the suggested edits)

You can use "Edit->Copy Mode" by default Shift+Cmd+C, In there you can use VIM-like shortcuts:

/ will allow you to search, v to select, And while selecting you can use $ to move to the end of the line and finally the good ol' Cmd + C to copy.