I'll just add something to what u/ribnag said: try cleaning the thermocouple. It's very common for them to get dirty due to years of a flame generating soot, and you want them to be clean to more accurately measure temperature.

So get a dollar bill and rub it. If you've seen The Big Lebowski, or are an old-timey serious bowler, you probably know how people polish their bowling balls by using a big cloth and pulling back and forth. Do the same thing using a dollar bill. You basically want to use it like sandpaper.

But it's important to know how it works:

• you turn the knob to pilot
• you hold down the knob, which releases gas out to the pilot flame
• you ignite the pilot by pushing that loud clicking button until a flame appears
• you hold it down for a while, which heats up the thermocouple
• when the thermocouple is sufficiently hot, it believes that it is detecting a flame (which it is)
• if the thermocouple gets too cool, it believes that no flame is present and shuts off the gas

So it's really important for that piece to stay clean, because the dirtier it is, the more heat is required to make it believe a flame is present. And since the pilot light is always uniform, it's possible it goes out because it's not detecting enough heat.