My wife and I and two kids stayed at Hotel Eiger for four nights in mid-August. It was probably the best place I've ever stayed. A good chunk of that credit goes to the town of Mürren and the amazing natural beauty that surrounds it, but the Eiger itself was flawless.
We were welcomed by the owner, although I didn't know that at the time and slipped him a few francs for showing us to our room, which had champagne and orange juice waiting for us. While we checked in, the owner made sure we were aware of the various sights and activities in the area, at the same time that our luggage was being taken directly from the train (the station is across the street from the Eiger's entrance) to our rooms.
We stayed in the Enzian Suite, which had two bedrooms and bathrooms, with a shared living room and balcony. The view from the balcony and from the living room, across the Lauterbrunnen valley, was absolutely stunning. Every time we returned to the room, I'd take a few more pictures, then sit and drink in the view. The lighting and fog or cloud cover continuously changed, so I never tired of staring at the mountains.
The wifi worked well everywhere we were, and it didn't require a password, which made it that much more convenient. The included breakfast was plentiful and delicious. Housekeeping was unobtrusive and pleasant.
We ate at the restaurant once, and the food was good but not great.
Mürren is extremely pleasant, with a variety of restaurants and shops, excellent access to nearby towns and the Schilthorn, and did I mention the great views?