You need to select the entire staff as usual, and then filter the selection
to contain only items in one voice. There are short cut keys for doing the
filtering. These are Ctrl+Alt+Shift+N, where N is the number of the voice. So,
Ctrl+Alt+Shift+2 will filter the selection so that it only contains voice 2
items.

The important thing to remember is that when you paste the notes, they will
keep there respective voice numbers. This means that if you want to copy
voice 2 notes onto a staff of there own, you may need to move those notes
into voice 1.

Example: copying voice 2 notes from a staff containing two voices onto a
staff of there own.

1. Go to the staff containing two voices.
2. Press Ctrl+Home then Ctrl+Shift+End to select the entire staff.
3. Press Ctrl+Alt+Shift+2 to select the voice 2 notes only.
4. Press Ctrl+C to copy these notes to the clipboard.
5. Press LeftArrow to clear the passage selection and return to a single
selection.
6. Go to the start of the staff that the selected notes will be placed on.
7. Press Ctrl+V to paste the notes.
8. Press Ctrl+Home and the Ctrl+Shift+End to select the entire staff.
9. Press Alt+1 to place everything on the staff in voice 1.

Some important things to note.

After select the first staff you must not forget to press Ctrl+Shift+Alt+2.
This ensures that the selection only contains voice 2 items; the voice 1
items are discarded from the selection.

After you've pasted the notes onto the new staff, you will actually have two
voices on the staff. This is because the staff already contains voice 1
items, even if they are only bar rests. The notes will still be in voice 2
but the bar rests which belong to voice 1 will still be there. Moreover the
notes will have stems pointing downwards which is not desirable.

Selecting the entire staff and pressing Alt+1 puts everything in voice 1.
Sibelius will attempt to merge the voices and because voice 1 only contains
rests, it will be completely replaced by the items in voice 2, which will
now be in voice 1.

Hope this isn't too confusing.