Heathrow Airport to Motel One tower hill

What time and what day of the week? How many people, how much luggage and what sort of fitness level? Makes a big difference to recommendations…

You will get a variety of answers here. I want to point out an app that will help you navigate London by public transport with ease. Use the CityMapper App. Put in current location, where you want to go and arrival or departure time. It will give you top options.

So that's on the Minories, between Fenchurch Street main line station and Aldgate East, which is one stop beyond Tower Hill towards Upminster so by tube it would be Piccadilly to Hammersmith or Earls Court, then District line. Much easier to navigate to from Aldgate East.

As isn31c says, except don't change at Earl's Court - change at either Hammersmith or Baron's Court, where it's just a cross platform change.
Another alternative would be the Elizabeth line from Heathrow to Whitechapel and then back one station on the District line to Aldgate East. Probably a bit quicker and more comfortable trains but a more awkward change at Whitechapel and a bit more expensive.

Any thoughts on getting from LGW airport to London? Instead of Heathrow ?

Take the Thameslink train to Blackfriars and change to the District line tube to Aldgate East. Try and sit at the front of the train to shorten your change distance.

Johnew52 is giving the best advice. Especially about changing at Hammersmith or Baron's Court. Do not change at Earls Court. It would be a big mistake.

But I'd like to step back a bit....what do you mean by "best?" Fastest? Cheapest? Easiest?