The timeline for when to restring begins with your choice of string. So let's look at the interplay of string selected, frequency of restringing and performance.
When should you restring?
Touring pros restring every day. Recreational players restring anywhere from every three or four times they play to once a decade, or until the strings break. But the pros' frequent restringing tells us something: String--especially fresh string--matters a lot. What possible difference could restringing your racket every day make?
Tennis string has an unfortunate property--beginning from the very second it is put into the racket, it loses tension. A racket strung at 60 pounds will most likely be at 50 pounds the next day, and tension continues to decline with every second and with every hit. Tension loss is the only physically significant process impacting your tennis racket (and string wear). This is why rackets need to be restrung.
To fully appreciate the benefits of fresh strings, you need to consider what tension loss does to racket performance.
More power, less control, change in stroke
As tension goes down, the strings stretch more upon impact. This cushions the ball's landing, minimizing the squashing effect. When the ball flattens, it loses a lot of energy. So less squashing means more energy for rebound. The strings always return almost all the energy that goes into stretching them, whatever the tension. So power is all about what happens to the ball, not what happens to the strings.
Power is good if you want it, can control it, and know how much to expect from day to day. That is how you groove your stroke--by responding the same way to the same situation. But your strings deliver varying amounts of power from day to day and from hit to hit. This works against grooving anything. As you struggle to keep the ball in, you constantly change your stroke.
More dwell time, less control, change in stroke
When the strings stretch more, the ball stays on the strings longer. The increase is only a millisecond or two (depending on where on the racket you hit and how violent the impact, dwell time is typically 5 to 7 milliseconds).
During that extra millisecond, your racket will sweep through both a larger vertical and horizontal arc. This will launch the ball on a higher and more sideways trajectory than you are used to. The ball goes long and wide.
This, coupled with more ball speed from less ball squashing, is a double whammy. You can't figure out what technical flaw has emerged in your stroke, and you begin to mess with perfectly good mechanics to fix your mysterious ailment.