Heathrow to Central London

A taxi will be convenient, but it will also be the most expensive way.

If you want door to door service, prebooking a car service will be less expensive than a taxi. Several people on this site have recommended Just Airports.

The cheapest way is to take the tube. Take the Piccadilly line to Hammersmith. At Hammersmith, switch to the District line (just across the platform) and continue to Victoria Station.

You can also take a National Express coach from the T3 bus station to Victoria Coach Station:

http://www.nationalexpress.com/routes/heathrow-to-london.aspx

This should be an easy walk to Ebury Street. Sit upstairs in front if they still use double-decker buses (they did when I used it 20 years ago, but maybe not any more).

Brave the tube, its really easy to use and people will be very happy to help if you get stuck. If you want the luxury of being dropped at your hotel door without the full expense of a cab from the airport... Take the Piccadilly line from T3 to Knightsbridge then exit the tube and hail any black cab to get dropped directly to the hotel door. Tube fare with Oyster is £3.10 each and a black cab the mile or two to the hotel maybe £10. Tube wont be busy around midday and no change of lines needed.

If you want a direct way with minimal effort involved due to luggage, then I would agree to pre-book a car as opposed to taking a taxi. A black cab from Heathrow to Central London will be much more expensive.

If you want fairly direct way and don't mind some transfer without too much lugging, then I'd take the Heathrow Express to Paddington, then a cab from there. That should get you there without having to lug bags up and down stairs and such, for much cheaper.

One note on the tube... it's certainly the cheapest way to go, but there are not elevators in most tube stations, so you will have do some lugging of bags up and down stairs at some point if you have to transfer lines. If you are traveling light, this would be a great option.

EDIT: Though, Mike has posted a way that would be even easier for you above so please take note of his research, which is much better than mine :)

Heathrow Express is a fine alternative :o) quicker, but a little more expensive. I always consider the tube first as i'm far more familiar/comfortable with it.

I agree with Lola. For where you are going in London, the bus from Heathrow would be a good option.
Take the National Express coach from the T3 bus station to Victoria Coach Station.

This is what we did last spring (may 2016) when we arrived at Heathrow. Just follow the crowd, upon deboarding,
and the signs in the airport will direct you to the bus station.

You will see the ticket counter for National Express. Just go to the window and purchase a ticket for the bus to Victoria Coach Station, London. It will be about 6 pounds. (If you are a senior, say so, and get a pound or two off.) The attendant at the window will tell you, "Bus 22" or whatever. Be specific about where you are buying a ticket to go to, because buses from this station go all over England. It may seem obvious to you that you want London, but be sure to speak up.

You will go outside some sliding glass double doors, find a seat on a bench, and watch for your bus 22 to pull into its parking place under the covered shed. Make sure the bus you're looking for isn't already sitting there.

When your bus arrives, get right over to it. They don't stay parked very long. The bus driver will be standing in front of his bus or beside its door. He will take any luggage you have that is larger than carry-on and put it in the (closed) storage bins underneath the bus.

When you get onboard, take one of the front seats if you can. The view is great through the huge front window of the bus as you ride into London. There is no double decker bus anymore on this route. There is a single floor. The seats are very comfortable, and nicer than airline seats. They have WiFi onboard.

You'll see the crowded motorways as you ride into London. Also some interesting Victorian-era warehouses and townhomes. At this point, the roadway is on an elevated bridge over the neighborhoods, so you can look down the streets of some very old neighborhoods.
As you get into London, you will enter Chelsea, now down on the surface roads. Look on the left side of the bus to see the beautiful pub "The Sun In Splendour".
In this neighborhood, you will also see mothers with strollers and babies on the sidewalk, people walking home with sacks of groceries, and luxury cars parallel parked on this busy street. All very interesting.

When you arrive at Victoria Coach Station, the bus driver will get your large bags out from the storage compartments. Be careful exiting the station. There are many buses pulling in very fast.

If you walk west out of the parking lot for Victoria Coach station, past the Shell station, you will be looking at Ebury Street. It runs north and south on the west side of the bus station.

I found this journey into London to be more interesting than riding the tube. And it puts you right where you need to be in London, with no extra taxi journey from train station to hotel.

If you got a lot of luggage and a lot of money take the taxi.

The potential problem with the bus is midday London traffic can be bad. I have only tried the bus from Stansted but it took about an hour longer than train plus cab. I may have just been unlucky. After a long transatlantic flight I don't feel like sitting on a bus for 1.5 hours or more.

For a pre-booked car service, Blackberry Cars are often recommended on TA:
http://www.blackberrycars.com/

I don't normally recommend the National Express coach mainly because of traffic issues although since you are in the unusual situation of being (possibly) within spitting distance of Victoria Coach station (only possibly because Ebury Street is fairly long) it may be for once a reasonable choice. Do be aware that these coaches are not a dedicated airport service, but long distance services to London that are routed via Heathrow.

Everyone is so helpful. I am carrying luggage so I will have one bag. I don't mind taking the tube but it makes me a little nervous as I am not familiar with it. Is it easy to find in T3 and are directions clear as to where I need to go? I don't want to head in the wrong direction. If I choose a cab can anyone estimate the price from LHR to Ebury Street? Thanks!!!

It is pretty easy to take the tube. But a Taxi will cost $75-$90.
Ask the hotel what to do.

The tube stations at Heathrow are easy to find, but if you're having trouble, you can ask anyone working in the airport to point you in the right direction.

Heathrow is the end of the line, so there's no danger of getting on a train going the wrong way. From Terminal 3, just avoid the trains that say they're going to Terminal 5, and you'll be good. (Alternatively: Follow everyone else - they're all heading toward Central London like you are.)

A black cab would be a good £60+ depending on traffic. You'll be amazed how easy the tube is, honestly. Even if you go the wrong way, just get off, cross to the other platform and come back. The whole network is very user friendly. Have fun with it... No one ever got on the tube and didn't pop out again somewhere :o)