Both are valid, but a bit awkward. They could be smoother, but are not incorrect. I would say: "On weekends, she joins her friends for a barbecue." This sounds slightly more cohesive to the native ear.

I would say both are correct.

The "grammar" is fine but I would question the intent. To meet to do X is short for To meet IN ORDER TO do X So is she meeting IN ORDER TO have a barbecue? Sounds strange. As if the bbq was the goal. Perhaps she is meeting her friends to have a good chat and catch. To meet for X Again this is about purpose and intent. To meet my friends for dinner. This is fine but it puts the focus on the dinner not the friends. Which is obviously possible. Come to my house for a barbecue. The barbecue is the "star". So in your sentence do you want to focus on meeting friends or having a barbecue? If it is the friends part then you could add the barbecue as extra info rather than the focus. eg On weekends she meets up with her friends and they (often) have* a barbecue. *other verbs possible: order, cook, do, organize etc

I say, "At the weekend/At weekends..." I'm very British (and OLD) so ON seems strange to me still! Ha ha! But I will try with ON... On weekends, she meets her friends FOR a BBQ. This seems good. They meet, somewhere. Probably not in her garden, but somewhere else. On weekends, she meets her friends to HAVE a BBQ. This is ok too but it doesn't seem very clear where. If you HAVE a BBQ then you are in charge and you cook the food. But MEET suggests she goes to meet her friends. So who is HAVING the BBQ? Where? Who is cooking? If it does not matter who or where, I'd use FOR.

In the U.S. the concept of "barbecue" has significant regional variations in meaning. The ingredients, the kind of meat, and the nature of the place where it is eaten are different in, say, the Southeast and the west coast. In different places, people will talk about barbecues differently. To me, in the southeast, it sounds natural to say "let's order some barbecue" because almost nobody makes their own. Do do so, you need special equipment. For us, "barbecue" is more a food than an event. It's something you eat. On the west coast, it is synonymous with "grilling".