It could be that this particular behaviour (button light turning on and off repeatedly, clicking sound, stopping after some time) means something specific and is described in the user manual, or on the lenovo site, or that their customer service can tell you what it means.
If no luck with that, it looks like a motherboard failure. If out of warranty, replacing it might be expensive.
I have a Lenovo laptop that I bought barely an year ago. Everything was fine until now, I was using it, doing regular stuff and then it just shut down all of a sudden. Now, every time when I press the power button, nothing shows up on the screen, but the light on the button turns on and off repeatedly and makes a clicking noise, then it stops at some point. I kept the laptop on charge for every time I turned it on, if it means something. So what's the problem? What can I do to fix it?