Probably not worth it in my own view, bmi's best routes, and the ones worth enjoying are the mid hauls, which need more than a day.

You could go to Edinburgh or Dublin for the day, which both tend to go down well with an American crowd but you will spend most of the day traveling to the airport and then to the city centre. At least the GUVs, if there is availability, will give you a sandwich... If you are really keen would be a return flight to Edinburgh to see the castle, which is nice and the airport isn't too far away from the city. This will however be expensive, potentially more than your transatlantic flight (unless you are in business or above) and you'd be better just starting in London imho. Domestic flights could be great if you could get a mid haul on domestic duty but on Saturday they are likely to be on contact elsewhere. I've had good luck with the mid hauls from Edinburgh on a Friday, as already mentioned - at weekends the service reduces and turns into tiny planes (fun in their own way...).

However, it may be with considering Berlin (as per some recent comments on a mileage run) as an alternative to a domestic. You get better, "proper" Intra-European business class (i.e. 2x2 on a standard plane with the middle seat blocked) and hot food, access to the international lounge and the airport is close to the city centre. Having said that, it is a two hour flight so you wouldn't have too much time in Berlin itself though the times would lend for a 7am departure from London and returning at 7pm so you could have a nice full day.