In the C language where did they come up with the name
It means Ascii to Integer. Likewise, you can have
atol for Ascii to Long,
atof for Ascii to Float, etc.
A Google search for 'atoi "ascii to integer"' confirms this on several pages.
atoi(III): convert ASCII to integer
So even if there isn't any documentation more official than man pages indicating that
atoi means Ascii to Integer (I suspect there is and I just haven't been able to locate it), it's been Ascii to Integer by convention at least since 1971.