The way the book’s protagonist views it, staying put is a valid strategy if you have the numbers. The Career kids from districts 1, 2, and 4, who trained their whole lives for this and almost always team up, constitute a full quarter of the total competitors, and after the bloodbath on the first day they usually constitute about half of the remaining kids. With that much force they can afford to set up a camp, since they’re reasonably likely to repel any kind of coordinated assault on their position.
As an individual or small group, however, you have to assume that groups and/or particularly deadly individuals are also constantly on the move, looking for survivors to gank. In that case staying in one place can be potentially fatal, as it vastly increases the chance of a systematic search finding you and gives a savvy individual player the chance to stumble onto your camp and set an ambush for when you return.
Book 1 spoilers:
There’s a point near the end when the protagonist teams up with her friend. His leg has been badly messed up by a sword, and he can’t really walk, so they end up setting up a permanent camp hidden in a small cave. Despite how well-concealed their camp is, they both consider how dangerous it is- essentially any one of the remaining tributes could’ve very easily rigged an ambush and ganked Katniss before she knew anything was up.
For my money if I was playing solo, I would never want to stay in one place- being constantly on the move means that as long as I keep my water and food topped off, I never have to fight someone on anything but my terms.