Best way to travel into London from Heathrow Airport?

Bu Oxford I assume you mean Oxford Circus and not the train station serving the city of Oxford?

If so, get the Tube.

Picadilly Line, the Heathrow Connect and Heathrow Express go to Paddington which isn't near.
Do you mean Oxford Circus tube station? If so, you'll have to transfer.

Here's a link to the numerous tube stations along Oxford Street:

https://www.oxfordstreet.co.uk/visiting/getting-here/

The Piccadilly Line does not go to Paddington, nor to Oxford Circus. Assuming you are headed to oxford Circus, transfer from the Piccadilly Line to the Victoria Line at Green Park, and go one stop. If you take either the Connect or the Express, you will need to transfer to the Underground to get to Oxford Circus....probably the Bakerloo Line going about 5 stops.

Oxford Circus? Take the Piccadilly line to Green Park. Transfer to Victoria Line and go one stop to Oxford Circus. You can plan your route on the TFL website: https://tfl.gov.uk/plan-a-journey/

The tube for sure.

Sorry! I'm staying in a hotel near Piccadilly Circus and Oxford Street/Oxford Circus area.
Will the tube get crowded? Is it pretty convenient?

I don't mind transferring though, but convenience is #1

Will the tube get crowded? Yes, but you will be getting on at the first or maybe second stop. Piccadilly Circus station would be a direct route on the Piccadilly Line, but does not have step free access to the street if this is important for you.

I take the Tube every time (except once I took Heathrow Express, and would not again). What time do you land? I usually land around 9:00 am and find the Tube is not crowded at all.

Since you're going to Oxford Circus, you do have one reasonable option to the Tube, and that is Heathrow Connect to Ealing Broadway, then Central Line to Oxford Circus. The only (slight) advantage to this is that you would be transferring with your luggage at Ealing Broadway (which is the terminus for the Central line and perhaps easier to get onto the train and find room for your luggage); as opposed to transferring at Green Park which is one of the busiest stations in London.

I'd still take the Tube, but if you have concerns about transferring with luggage in a busy station, then consider the Heathrow Connect / Central Line option.

Just noticed your hotel is near Piccadilly Circus? If so, forget what I said earlier. Just get on the Piccadilly Line at Heathrow and stay on right through to Piccadilly Circus.

Piccadilly Circus and Oxford St. Station are slightly over half a mile apart. Please tell us either the name of the hotel or address so we can give you better directions.

Flying in in the morning could mean 6am - which could mean reaching central London around 8 or 9 and it will be crowded on a weekday and less so on a weekend - or it could be 11 am with you going through central London maybe at 2 in the afternoon when the tube will be less crowded. But the tube is never really un-crowded.

Several times you've mentioned convenience and comfort and also referenced the "best" way. You've also mentioned you would take those over cheapness. That being the case, the best, most comfortable, and most convenient way is a private car service. In a brand new place there isn't anything better than having a private driver meet you inside the terminal, assist you with luggage, and whisk you to your hotel. No, it's not the cheapest way, but definitely the best. Prices vary widely so get a number of quotes if you decide to use a car service.

Hotel: Sanctum Soho Hotel
Arrival: Sunday at 9:00 AM at Heathrow Airport

I'm bringing a small luggage and a backpack with me. I don't mind transferring as long as it's nothing confusing. Tube sounds fine. I've taken trains and subways in Tokyo during rush hour, which is a nightmare, so I think I can survive.

Actually that hotel is for a later day.
Sorry, I'm staying at the YHA London Oxford Street, 14 Noel Street on the first day!

Just to point out that you can buy online a ticket on the Heathrow Express 90 days or longer ahead at a very low cost. It takes about 20 minutes to Paddington, from where you can take the Bakerloo line or a taxi if you prefer. The Heathrow Express ticket commits you to a particular day but not a particular train, so there's no problem if your flight is late or immigration takes too long.

I just tonight booked a round-trip private transfer with "Twelve Transfers", for May. We will be picked up at Heathrow T5 and taken to our London Conrad St. James. Then we will do the return transfer also. It gets high reviews on Trip Advisor. My husband and I have always preferred to have a private transfer. We like being met where we pick up our luggage. We don't want to hassle with our luggage.

Emma's plan will work perfectly. With one small luggage and a backpack it should be no trouble on the escalators. Easy, not particularly slow, very cheap.

I think I'll try out Emma's suggestion!
Thanks everyone! All your inputs really helped :)