What is the ruling on my mother teaching English outside of official working hours to students of a family whose source of wealth is not known with certainty, and is it obligatory for her to verify the source of the wealth?

The default assumption regarding a Muslim's wealth is that it is lawful, and there is no obligation to investigate its source. Therefore, there is no harm in accepting payment from this man's money unless it is proven that all of his wealth is unlawful. If it is proven that some of his wealth is unlawful, then accepting payment from him is not ruled as unlawful, but merely disliked. This is because dealing with and benefiting from the wealth of a person whose money is mixed (with unlawful elements) is not forbidden, but only disliked.