You have to have a game in the system for it to show up on the screen there is no splash screen like later consoles. It may be a bad a/v or rf cable depending on what you are using. Honestly your best bet is to find someone with a snes and borrow their cables to test the system itself. That way you can tell if it is your cables or the system. A friendly game store may help you test it if you were to purchase a replacement cable from them if that proved to be the case.
I know this post is super old but I had the same issue and I found a fix. Turns out the power cord was bad even though the little red power light went on. I must have been getting enough power for the light but not enough to turn on the system. Hope this helps somebody.
Stupid answer but before you go tearing your snes apart check the power cord. I've had a few die on me recently they are getting pretty old. One of them even still lit up the red power light but wouldnt power the system.