Suggestion from outside the box but....
How about something from London? e.g. New York
Reasons - Currently QF1 and QF9 from Melbourne and Sydney arrive at Heathrow at roughly 5.30am and 6.30am (depending on time of year) and sit idle for 15 / 17 hrs until QF2 and QF10 leave late at night.
Pretty sure these planes aren't cheap and to have them sat around for 15/17 hrs every day for 365 days of the year is not a great use of resources. Now New York may not be the best destination but at least it could then connect up with QF107 / QF108 for those RTW types and Business Travellers. Could even make the trip and back in a day and still be there for the night flights back to Oz.
To say the QFi isn't making money but then to have 2 expensive planes sitting around all day making $0 seems ridiculous.
Interesting comment about the A380s sitting idle for so long. Just a thought...
As it stands, an average A380 would route something like this over 6 days:
Day 1: QF1 SYD 15:50 - DXB 12:35 (+1)
Day 2: QF1 DXB 02:05 - LHR 06:35
Day 2: QF10 LHR 22:30 - DXB 08:20 (+1)
Day 3: QF10 DXB 09:50 - MEL 05:15 (+1)
Day 4: QF93 MEL 09:15 - LAX 06:35
Day 4: QF12 LAX 22:10 - SYD 06:10 (+2)
Day 6: Back to LHR (or perhaps HKG)
Of course, this could vary and the plane might take a day "off" this cycle to go to HKG and back from SYD, but this is quite a common routing.
If QF retimed it's flights, this would be possible in one day less per cycle, for example:
Day 1: QF1 SYD 21:30 - DXB 06:15 (+1)
Day 2: QF1 DXB 07:45 - LHR 12:15
Day 2: QF10 LHR 22:30 - DXB 08:20 (+1)
Day 3: QF10 DXB 09:50 - MEL 05:15 (+1)
Day 4: QF93 MEL 09:15 - LAX 06:35
Day 4: QF12 LAX 09:10 - SYD 17:10 (+1)
Day 5: Back to LHR (or HKG)
This would cut out a lot of the time the aircraft spend idle in both LHR and LAX (while still giving it sufficient time for maintenance, cleaning etc. especially in LHR where it would still spend 10 hours +. The timing of the DXB-LHR flight would also be a lot more convenient for people just wanting to fly this leg (I booked this route recently but ended up booking with Emirates at a different time because I didn't want to get a flight which leaves at 2am).
By saving a day every 6 days, QF could also fly more routes with the same number of aircraft, e.g. the HKG route, which takes a day's cycle out of SYD. QF could add another such route, e.g. they could fly an A380 SYD-SIN (and free up one of the planes which currently flies to SIN so as to add a new route, or perhaps re-open an old one) or they could even open a new route! I know that's not something QF management is familiar with, but maybe there's some hope! If each of the 12 A380s "gained" even just one day a week, that's 12 extra days available. If a trip, say, from SYD-YVR-SYD lasted 2 days (as per current time tabling), then QF could fly this route 6x weekly with A380s without needing any new aircraft!
Why not do this? Well, quite a few of the flights are timed as they are so they connect with other services. It would also take away time for maintenance etc. (although I do wonder how long the planes are just sitting there...) I'm sure there are probably a few more other reasons, but do they outweigh the benefits?