I will leave you to your own devices for getting a passport and sorting out your flight details. But one of the most important parts about your stay in London? The place you'll rest your weary feet at the end of touristing.
I am incredibly cheap when it comes to hotel-age. When I visit London, I usually have the luxury of crashing on people's couches. But failing that, I will stay in a hostel (or a hotel if someone else is paying). Though I'm kind of a lazy tourist, I do not spend too terribly much time in the bed wherever it is I'm staying, so I don't typically choose luxury accomodations. I look for safety, location and price.
With that in mind, here are two standbys for trips to Londinium:
London School of Economics Vacation Accomodations
This is definitely my top choice when travelling to London. I stayed in Passfield Hall, located just off Tottenham Court Road and a short walk from two tube stations (Warren Street and Euston Square, both short rides to the Picadilly Line, which will take you to Heathrow). Reasonably priced, safe area, free breakfast, and if you're a couple, you can stay in the same room. The rooms are pretty bare bones, a bed, a sink, a desk, a phone. Think the inside of a dorm room before it's furnished (they do provide towels and blankets/pillows, as well as daily linen changing). You will need to share a bathroom. There's a common room with a TV, books, etc. But you didn't visit London to stay inside all day! There are several internet cafes and a grocery store nearby, and you're about a 15-20 minute walk from Oxford Street (I need to quit talking about it, it makes me want to shop). The only downside is that many of the halls double as residence halls, so you may only be able to book accomodation during school holidays.
London House Hotel
This is a really awesome location in Bayswater. The rooms are a decent rate (can get pricey), but if you're a couple, you will be able to share a room here. There is a TV in each room as well as an ensuite bathroom. The Tube station is just across the road and there are a ton of great pubs nearby. Negatives for me at this hotel? The woman who checked me in had a very loose grasp on the English language and there is no elevator. But if you don't have much luggage, shouldn't be a problem. This place books up quickly, so book well in advance.
If you're travelling solo and want to meet people (or if you are travelling in a group and can book enough beds in a room to get it all to yourself), I recommend St. Christopher's Inn in Hammersmith. Located directly across the street from a Tube station, above a great, cheap pub (and you get discounts for staying at the hostel) that has live music some nights. Free wi-fi in the pub, and actually really comfy beds. I have stayed here a number of times and I absolutely loved it (plus I met my London Partner in Crime, Steph, at this hostel!).