What is the origin of the Hard Rock Cafe name?

the origin of the name is because it is a cafe that likes hard rock.
http://www.hardrock.com/corporate/history/

The first Hard Rock Cafe (HRC) opened its doors to the public on June 14, 1971, in London, England. Founded by Isaac Tigrett and Peter Morton, two enterprising and music-loving Americans, HRC was an instant classic, attracting droves of customers with its first-rate, but moderately priced casual American fare, warm service and ever-present rock 'n' roll music and sensibility. Although it was initially decorated with an eye toward eclectic Americana, HRC has ultimately become the world's leading collector and exhibitors of rock 'n' roll memorabilia.

Further research has led me to believe that it was from a Doors song either entitled Hard Rock Cafe or containing the words in a lyric. Any further comment?

The Doors' "Morrison Hotel" album cover artwork consists of the band members standing behind a plate glass window of the "Morrison Hotel" ("Rooms For Rent"). Nothing else. However, the Doors gave a name to each side of the record. On the back of the cover, side two is sub-headed "Morrison Hotel," while side one is simply entitled "Hard Rock Cafe." Otherwise, there are no songs or lyrics on the album that use the words "Hard Rock Cafe."

Did this influence the men who founded the Hard Rock Cafe? Yes, according to the guy who took the picture, rock photographer Henry Diltz. In a 1997 interview with radio personality Paul Harris, Diltz tells how in 1969 he took the group's picture at the seedy Morrison Hotel--yes, it really existed--in downtown L.A. Looking to grab a beer afterward, the group was driving through L.A.'s skid row when someone spotted the Hard Rock Cafe. Of course they had to stop. Having tossed back a few with the local winos, they posed for the picture that wound up on the album.
Says Diltz:
I guess though sometime the next year after the album came out with that picture on the back, they got a call from England and this guy says, "Hello. Would you mind if we use that name on the back of your album? We're starting a cafe over here in London and we would like to use that name." And they said no, go ahead, and that was the beginning of it. Now every time I go into a Hard Rock Cafe, whatever city I'm in, I always feel like I should get a free hamburger.
https://selvedgeyard.com/2011/03/02/the-doors-when-the-musics-over-morrison-hotel-hard-rock-cafe/