The main factors to consider are how long he/she/they will be in London, location of the accommodation (unless you are tasked with finding that too, eek!), interests and how much money they want/have to spend.
First they need to get a guidebook and start visiting websites like Rick Steves. There is a helpline that can answer all sorts of questions and relieve you of the burden. I like TimeOut too for ideas. London is HUGE. Even after five years, I am still finding new things.
London is 'exotic' but easy to figure out because it is all in English and people will help. It isn't that scary if you show basic confidence. There are tourists everywhere.
The no-brainer London trip is the on-off sightseeing buses. There are at least three companies and pretty much offer the same. Big Bus Tours, The Original London Sightseeing Tour and CitySightseeing. Check for the free Thames tour.
I highly recommend it to give a good idea of the spread of London. If the weather is bad or they just have a couple of days they can hit the main spots with little effort. Prices are reasonable too.
London is very walkable once you know your way around. Walking tours are good too. Self-guided with a map or download a podcast or join a walking tour. There is even a 'free' walking tour that you only pay tips. I've been on few in Berlin, Dublin and London and always an excellent value. Sandeman's New Europe: London
There are so many museums but the biggie is the British Museum for a taster but if they like to museum hop there is also the V&A, Natural History Museum and Science Museum all on the same block. Plus, dozens of other free or reasonably priced museums dotted around. London sightseeing can be done on the cheap or big bucks of they want to see the 'touristy' things that have entrance fees.
Oh and markets!! Don't get me started! A Sunday over in Spittalfields and Brick Lane area or a weekend day in Greenwich (or Greenwich in general) or everyday Covent Garden. There are so many to name.
A night out at the theatre is very doable and some great deals can be had by booking online. Lastminute.com does theatre and meal deals and LondonTown is good for cheap tickets. I've never used the 1/2 price ticket booths in London but if they have the time and money it might be worth it if they want to pick on the day. Likewise, going straight to a ticket office of a show they want can get them seats in their price range.
http://www.londonforfree.net/index.phpMy advice is to point them into the direction of experts and save yourself the grief.
Any London airport will do. Transport links are fabulous on most days.