From Heathrow you're on the piccadilly line, which is what most attractions are dotted along.
I'd start by jumping off at south kensington and checking out the natural history museum, even if it's just to look at the building and take some snaps, it's pretty impressive.
All museums are free here btw, so feel free to run in and look at dinosaurs and other, natural sh*t.
Get back on the piccadilly line, get off at Knightsbridge and checkout Harrods and 1 Hyde Park corner (Hyde park's big, you won't have time to see much of it but it's worth a gander).
Get back on the piccadilly line and get off at piccadilly circus (you can also get off at green park instead and walk down piccadilly towards the circus). That's the famous circus, it's a bit like NYs time square kinda with all the advertising signs.
Checkout piccadilly circus and leicester square, if you can, walk across to covent garden too (ask for directions, you don't need to get the tube but if you're uncomfortable working out the way [London's has LOTS of roads and they are usually very confusing for my American mates because they're used to grids, we have zero, grids] then get back on the tube to covent garden).
Covent garden is awesome, lots of nice restaurants, walk south onto the strand and back up to trafalgar square to see nelson's column, if you want you can checkout 'The Mall' which is the road that leads from Trafalgar square to Buckingham palace.
From trafalgar square the mall takes you to buckingham palace but I'd probably walk down to the river and follow it, you'll be opposite waterloo and south bank (checkout south bank, it's a huge arts centre and it's where a lot of galleries are).
To the right from embankment you've got Big Ben and the houses of parliament, you may be a little surprised to see Big Ben isn't that big now compared to other buildings, you can actually go up inside it but you probably won't have time.
Like I said screw the London eye and if you can *book* Oblix Bar at the shard (you will absolutely need to book) and use the shard as a viewpoint instead, it's impressive and lets you see the city in all directions and all its glory, from above.
Use the Thames Clippers (boat service) once you've gotten to the river part (starting at embankment), there's piers at westminster (houses of parliament / big ben), embankment (trafalgar square / the mall and bucks palace), london bridge (the shard), tower bridge (tower bridge, tower of London), the clippers aren't the cheapest but it's by far the best way to get from attraction to attraction for the things that are on the river.
There's so much to do and see I couldn't even start to list everything here, if you want anymore tips just ask, heck if you're spending the day here on a Saturday I'll even happily meet up and give you a tour of shit you don't get to see as a tourist.