What do you put in child themes author tag?

There are several scenarios:

1. If you had authored your child theme from scratch, using only your custom code then it is clearly your product.
2. If you had used code from parent theme in your child theme then it’s your derivative of it, it’s still yours but you also need to credit code used in line with its licensing requirements.
3. If you had taken third party child theme and modified it you usually choose between keeping authorship and noting your modifications (usually for private or semi–private use) or renaming it and treating it as derivative same as 2 (usually if you intend to distribute publicly).

In either case pay close attention to licensing requirements of original theme.