First of all, Etymonline agrees:
1909, from common pronunciation of "that's the boy!" a cheer of encouragement or approval.
Merriam-Webster throws in a "probably" for good measure:
probably alteration of that's the boy
First Known Use: 1909
The most extended discussion I was able to find is in the Dictionary of Slang and Unconventional English by Eric Partridge:
attaboy! Go it!: US (— 1917); anglicised in 1918. (F. & G.) The OED and Collinson
derive it from that's the boy!, but possibly it represents at her, boy!, where
her is sexless; probably, however, it is a corruption of the exclamatory US staboy
recorded by Thornton. Dr Douglas Leechman, that eminent anthropologist and notable contributor
to the Dictionary of Canadian English, wrote to me in 1969: 'Everybody, except the pundits,
knows that this is "That's the boy"—"'at's a boy"—"atta boy".'—2. Hence, an approbatory
exclam. from ca. 1931, as in D.L. Sayers, Murder Must Advertise, 1933, '"Picture of
nice girl bending down to put the cushion in the corner of the [railway] carriage. And
the headline [of the advertistement]? 'Don't let them pinch your seat.'" "Attaboy!"
said Mr Bredon [Lord Peter Wimsey].'—3. (As Attaboy) an Air Transport Auxiliary
'plane or member: WW2, then nostalgic. (Jackson.) Suggested by the intitials and punning
on senses 1 and 2. See Ancient and Tattered Airmen or Aviators.