Where did the name `atoi` come from?

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.

I'm having trouble finding any official source on it... but in this listing of man pages from Third Edition Unix (1973) collected by Dennis Ritchie himself, it does contain the line:

atoi(III): convert ASCII to integer

In fact, even the first edition Unix (ca 1971) man pages list atoi as meaning 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.