What is the best value for accommodation on the Zambian side and nearest to the falls?
The closest by four or five kilometers are the Royal Livingstone and the Avani, the Avani being the cheapest of the two. The problem is when you say "best value". It's a subjective opinion rather than objective. You'd probably be best looking at reviews and seeing what suits you personally and what your budget is.
As Mark says, the two closest to the Falls are not cheap. Other than these two, anywhere in Livingstone will serve you well and provide some cheaper opportunities among them.
We stayed at the Stanley Safari Lodge that has a great view from the hill bordering the Mosi-oa-Tunya National Park. You can also see the Spray of the Victoria falls and hear the roaring of the lions from the nearby sanctuary. The food and service is great. It was not super cheap but it was a great option for us. Make sure you get the Kaza visa (same price than the Zambian 's Visa) so you can see the falls from both side (Zambia and Zimbabwe).
Avani is brilliantly situated (but pricey). Room rate includes unlimited entrance to the Falls and you can stroll there, so no taxi fees.
I stayed at Jolly Boys --quite good actualy