What made 800 feet and later 1,000 feet attractive as the standard traffic pattern altitude?
The traffic pattern altitude (and the pattern turn direction) is set by the airport owner in coordination with the FAA. Refer to section 8.

Airport traffic patterns are often 1000 ft. AGL today. In recent past, the typical altitude above ground level was 800 ft. Flying Magazine, in a September 1958 article states:


  Under VFR conditions proper entry into the zone is sadly neglected. Before you enter the control zone you should be at the Pattern Altitude which is more or less standard rule at every airport in the country--800 to 1,000 feet for light aircraft, 1,200 to 1,500 for heavy or fast aircraft. This provides you with altitude, yet gives plenty of room to make your turns. Letting down in traffic makes you vulnerable and blind, at the same time increasing the odds of hitting someone you will never see"


Experts and non-experts seem to agree an altitude increase would be beneficial:


Better noise abatement near the airport environment
Increasing tower heights
1000 is easier to remember than 800


Fun facts:


Tangier island/KTGI airport elevation is 6 ft mean sea level. The traffic pattern altitude is 605 ft.
Truckee/KTRK is 5901 ft. MSL with an 1100 ft. TPA.
Sedona/KSEZ is 4830 ft. MSL. The TPA is 1173 ft.


Pilots flying into Sedona must be precise, haha!