Hi headsouth, nice to hear from another EVA flyer!
First off, although right now they're flying 777-300 on this route, at the start of July this will switch to 747 for the peak season.
Some of the best seats are kept held back for frequent flyers (FFs) - ie holders of EVA's Diamond, Gold and maybe Silver card. If you have one of these, you should phone an EVA office and mention it when you book the seat. If you book online, they will be blocked off - even though the system "knows" you.
Sometimes seats are blocked off from the online system when really they are available to allcomers. So if you're struggling its worth phoning them.
You can find seat maps on EVA's site.
For 777, these are the best seats, in my opinion:
60A, 60K (exit window)
45A, 45K (exit window)
60C, 60H (exit aisle)
45C, 45H (exit aisle)
I prefer 60 to 45, but there's not much to choose. NB, the emergency chute doesn't interfere with leg room on the 777. You have great leg room on these seats! BUT, they are only pre-bookable if you are gold/diamond, and even that can be a struggle. Because of this, they may be available at check-in... so do ask. When I've been in them, the person I've been next to always got it at check-in. They weren't FFs.
Else, I would go for an aisle as near to the back as possible. 69-71 C/H are good choices - as the configuration is 2-3-2 here, instead of 3-3-3. If the flight is not full, you're in a good position to move quickly - the back seems to be last to fill up. Or you could go for a D/G and hope that the middle seat (E) is empty. Again, you can ask at check-in, they are quite helpful at LHR.
For 747
This is an unusual configuration - apart from being in rows 45 to 70 at the back, the upper deck (80-89) is also economy - but with a generous 36" pitch! Again these only seem bookable for FFs.
The best seats in my opinion (and in this order):
85K (exit window) - and that's where I'll be on Thursday!
85C/H (exit aisle)
85A (exit window) - but a slight obstruction from the door
Then you have the exit aisles downstairs - 45/55 C/H. I think the downstairs exits windows are obstructed by the chute.. avoid.
Then I'd go for any aisle upstairs.... maybe window too if you like. You get a locker by the windows left over from when this was C/F.
Then I'd go for a centre aisle as near to the back downstairs as possible (ie 70D/G, 69D/G) etc, hoping you wont get someone next to you, and perhaps a whole row free.
As I say, many of these seats will be blocked off unless you're a FF and phone them. BUT, its well worth asking at check-in, especially if you arrive early. If you're not a member of EVA's club and think you might make this journey more than a couple of times then its well worth joining. Although not as generous as before, unless you are on heavily discounted tickets, you don't need many trips to get a Silver card which would get you into the lounge at BKK and a few other perks.
Gus is the guru on economy seats on this route.. hopefully he will chip in with his opinions.