Is it permissible for an H1B holder to have extra income from home country (through freelance translating over the internet)

It would be risky. There is a possibility that USCIS would construe that as unauthorized employment as the work is being done in the US.

Risky, as the work would be performed here in the US.

I'm with my colleagues - there may be a legal argument that it's OK since there is no U.S. employer benefiting from the work, but you would need to list the income on tax returns and list the work on biographical info forms for any future green card case, and I think you would have a tough time explaining it as a practical matter.

If you perform the work while in the U.S., you would be engaged in unauthorized employment.