This one is always confusing. It should be simple, but it isn't. Not unless both parties mean the same thing by what they say. In the context, 'next' could mean,
1. The next weekend there is. (i.e. the next time it becomes a Saturday and then a Sunday)
2. A way of saying not this coming weekend, but the one after (or, the next one).
In another context, this would be nonsense. For example, if I said I would get the next bus, I wouldn't mean I'd miss the actual next one that came along and then board the one after! That would be stupid. But if someone saw a bus coming up the street and asked, 'Is this the bus you need to get?' and I said, 'No, I want the next one', we'd all know I'd mean that I intended to let this bus go past and get on the one after it. I guess it's all because we use the same words in different contexts instead of choosing a different word.
I often say something like, 'So not the weekend coming up, but the one after? Right?' Or, 'So not tomorrow, or the next day, or the next, but the day after that.'
A bit clarification can only be a good thing when arranging dates and times for things.