London Sightseeing Strategy

London is fascinating and 8 days is a good amount of time! Even then, you won't see everything on your list.
How to plan a London trip is a matter of personal preference, other here may well have a different view than mine.

Having said this, in general, your 2nd option may be best, your first option possibly may be more trouble planning than the value it gives you. Again, just my personal preference.
But....and just trying to be helpful here, let me respectfully suggest that you might be slightly over-thinking this. Planning is good, but over-planning sometimes backfires, IMHO.
Just one example: Weather and closure days can affect the most carefully laid plan.
And note that many/most London "sights" close by 6pm.

You'll likely be traveling back to your lodging on the Tube, so fatigue, though a factor, isn't going to be as big a factor as if you were walking back to the hotel every night.
The usual tourist sights in London are spread over an enormous geographical area, which is why you'll want to take the Tube or buses, or even taxis occasionally, to cover longer distances and save your energy for walking the sights and between sights located relatively close to each other.
You'll probably want to mostly take the Tube in Zones 1 and 2, these zones are covered by the tube passes and include many of the things you'll want to see, except outlying locations like Greenwich or Kew Gardens. Just try to avoid the Tube during commute hours.

Happy Travels!

Kent is absolutely right , you are over planning . One point to add - If you have chosen a destination for a particular day that is largely outdoors oriented , like roaming about in Hampstead , and it's raining , you want to be flexible and substitute an indoor site , like a museum , that day . Over planning creates rigidity and you really don't want ( or need ) that in a place with so many choices , like London .

Mikey,

what i did was more or less a combo of both.

I mapped all of the things i wanted to do/see and grouped them into areas. Then i went to the furthermost out attraction if it worked out time wise and waiting-line wise and then walked my way back from attraction to attraction. that way i could see the things up top.

happy trails.

I usually do the second option.

I agree with everyone else, try to group the sights geographically to save travel time. Fanning out from the hotel each day in a different direction and working your way back might be efficient if you're walking, but the city's too spread out for that. Better to walk, when possible, between sights in the same part of town and use tube and/or buses for longer distances. And be flexible in case of bad weather or unexpected delays.

I do the second option.

I spend rainy days at the museums. Do your outdoor activities first, such as a walking tour, enjoying the parks, open-top bus tour or taking a boat ride to Greenwich. Because you may have a rainy day or two that drives you inside. A rainy afternoon is also a good time to enjoy a pub lunch and drink.

And if you're really worried about getting tired, maybe you should prioritize your sights and do your "must-see" items of the day first (adjusted for weather changes) -- we found that museum visits worked best in the morning when we were alert.

Many of my London days end up being spent in a couple of fairly distinct parts of town. I usually start each day with a firm plan for the first site I'm heading to, from among the top priorities on my sightseeing list, that I don't want to miss. The middle of the day tends to be when my plans are more fluid to start with, and end up changing as I wander, or my tired feet decide they need a break. Then, I may have a firm evening plan, such as a theater ticket or special dinner reservation, and the location of those usually has nothing to do with my hotel. In smaller towns, it can be convenient to go back to your hotel before dinner, but in London, I expect to be gone all day, about 12 hours. If you're out so late that the Tube has stopped running, then there are some night buses, but you might decide that a taxi is a good value to get you home quickly.

I do what Laura (of RSE) does, in London.

When I visit London I usually have a couple of things I really want to see/do. I make sure I do one of those, then walk around whatever neighborhood I'm in, have some sort of lunch (usually a sandwich "on the fly"), then maybe head for another site. I sometimes plan to go for tea someplace. One day last fall I headed to St Paul's, walked around, saw a bus headed to the Tower. I was there to see the poppies in the moat at the Tower. I realized how logical it would be to catch a boat down to Greenwich from there, so that is what I did. I walked around there, bought a sandwich and found a riverside bench for lunch; after that I walked back to the area of the boat, visited the maritime museum for a while and took the boat back to the Tower. I was tired then, so I caught a cab back to my hotel. I often find myself at Trafalgar Square, enjoying the people, museums, St-Martins-in-the-Fields church for lunch. I may walk to Covent Garden from there. or Buckingham Palace. I almost never go out at night, especially when I am traveling alone.

Don't over plan. Weather may cause adjustments. Flexibility is key.

How far outside the city core are you thinking? Kew Gardens? Greenwich? If yes then I'd opt for #1. Otherwise you'll find a majority of the sites close to one another and therefore easy enjoy and explore in a timely manner. The tube can whisk you from one part of the city to another in a manner of minutes. At a tube station walk over to the manned ticket booth and look for the small pocket sized tube maps that are usually on the counters. If not there ask for one. VERY helpful to carry. Helps you see what tube stations are near your destinations.

For instance one day with an early start you could visit the Tower of London and then walk up to Leadenhall Market. Light meal or coffee then a few blocks walk to the Bank Tube station. Hope on Northern train (Black line on the map) and take it under the Thames to London Bridge. Mind the gap, exit the station and take in a view of London from the Shard. From there it's a short walk to Borough Market. Then follow Park Street over to Vinopolis, continue past it to the Thames and the Bankside development. From there easy stroll to the Shakespeare Globe and Tate Museum. Then walk across the Thames via the Millennium Bridge and on up to St Paul's Church. All doable depending on your stamina and time you spend at each destination.

You know your tolerance for walking, museum appreciation, etc. If you need to rest a nice pub or museum cafe lunch is a perfect means of doing so. Costa and Nero Coffees are plentiful (as are Starbucks). If you're traveling at a time of year when you'll see sunshine then by all means picnic in one of the many lovely parks. Lastly, if you enjoy seeing theatre then you must see it in London. Half Price ticket booth in Leicester Square is a great way to get same day tickets. You can now see what will be on sale online.

http://www.tkts.co.uk

Theatre is a personal preference when I'm in London so I base my days adventures based on the fact I have tickets for a performance that evening. Always allow for a respite before an early dinner and the play.

All about pacing yourselves. Don't rush. With 8 days you'll have plenty of time to appreciate London.