I'm not sure about 1977. The first chips that come up to me are the TI TMS9919 aka SN76489 and General Instruments AY-3-8910. The AY-3-8910 was around for sure in 1978. Both were also used in a lot of arcade machines and consoles/home computers. And both have follow-up designs that are available even today.
The direct predecessor of the TMS9919 was the SN76477, called the Complex Sound Generator (the TMS9919 was called the Digital Complex Sound Generator). Each function could be controlled via direct inputs, so clearly a sound chip. But for use with digital systems, some glue logic (latches) would be needed. It got an article in Popular Electronics in 1978 but I don't know how long it was before it was available.
On the GI side, there were the AY-1-0212 and AY-3-214 before the AY-3-8910. But it's hard to qualify them already as sound chips, as they weren't really self-sufficient from today's view.
While the Signetics 2637 isn't 'just' a sound chip, it offered in 1977 a single sound chanel plus white noise and loudness controll. The 2637 could be best described as a single chip video console, as it includes a video circuitry with colour abilitiy and sprites, 13x16 text, character ROM plus definable characters plus ADC for up to 4 analoge paddles or two analogue joysticks - plus said sound generator.