Do I prebook a taxi from Heathrow to London King's Cross?

At the risk of repetition: Picadilly line tube is by far the cheapest option, requires no changes and at peak times is likely to be much faster. The roads into London can be very, very slow in the morning rush hour. As the trains start at Heathrow you should have no trouble getting a seat.

Taxis: be aware of one important thing, there is no designated pick-up area for booked taxis, taxis (and indeed private cars) are not permitted to stop at the drop-off areas and wait for customers. This rule is enforced with increased rigour these days. Hence your choices are:

a). Use the standard Black Taxi service. Orderly queues, regulated cabs, drivers have “the knowledge” and so know the routes and hotels; this is relatively expensive.

b). Book a taxi through one of many cab services, but verify how you will locate your driver. I’d advise against using any firm that will not have you met at the exit from the customs hall. Some firms may try to save parking fees by asking you to go up to the drop off areas, but you can easily end up playing dodge-the-policeman and waiting for your driver. This is very annoying when you are tired after a long flight. Your driver should pay for parking and either come to meet you (by far my preferred option) or state some very well-defined meeting point in the car park. You should expect to have the parking fee (£5 or more) added to your fare. All the cab firms should take your flight details and adjust their arrival times according to your actual schedule. You really need to have a UK cell-phone available as your driver may have trouble finding you.

I’ve never taken a taxi from LHR to London/King’s Cross. Picadilly is the way to go and I agree with others that’s it’s relatively easy as tube basically starts at LHR so it’s fairly empty. I’ve also taken Heathrow Express which is faster. Taxi is the last resort, more expensive and can take longer due to traffic.

If you haven’t already, do consider public transport options which are very good and start much cheaper. If you take Heathrow Express then the tube, it’s likely to be a little faster than a taxi; if you take the Piccadilly Line, as well as being very cheap, you don’t need to change trains so it’s still quite convenient (and might even be faster than a taxi if traffic is bad).

If you don’t mind paying extra and want a taxi for the convenience, and don’t mind risking being stuck in traffic:

It’s definitely cheaper to pre-book. Much cheaper. Simple economics: if you just turn up, you’re a captive market, if you shop around, there’s competition. London is a monster that lives and thrives on eating the wallets of people who didn’t plan ahead; if you do plan ahead though, it can be surprisingly cheap.

The first private hire result from googling “Heathrow taxis” gives a quote for £51. I won’t link to it because I recommend shopping around and checking reviews, it’s just to give you an idea. Uber starts even cheaper – estimates start at £33. Important note: locating your private hire driver might not be as straightforward at Heathrow as it could be due to Heathrow’s rules, see djna’s answer on this page for details.

I’ve never taken a just-turn-up taxi from a London airport but an answer here estimates them at £80 and I’m surprised at how cheap that is. TaxiFareFinder estimates it at between £68.83 and £122.07 depending on traffic, and I’d expect it to be towards the upper end given how congested that route tends to be. That said, there will definitely be a black cab in the taxi rank (there’s always a small risk a private hire car might be late, fail to arrive, etc), and for the extra money you do get convenience and a driver who’s passed “the knowledge” test and knows a heck of a lot about London (a private hire driver might be a new arrival following a satnav).

I notice you mention King’s Cross – if you’re taking a train, intercity trains in the UK are also usually much cheaper if you book them a few days in advance. It depends on the route, some it doesn’t make much or sometimes any difference at all, but for others (particularly trains up north from King’s Cross to Newcastle, Edinburgh etc) booking ahead can save you 75% or more. It’s sometimes also a bit of a lottery depending on how many tickets are sold, so get in early if it’s a long journey.