Relays are responsible for powering various systems in the vehicle. If there was a problem, there would be systems in the vehicle that would fail to activate. For any safety related systems, such as airbags, this would trigger dash warning lights and codes. For other systems, such as the radio, HVAC or engine, the results would be obvious.
I can provide you one example where the relay flicking on, then off, then back on is normal. For instance, you hop in, shut the door, hit the Start button, the fogs come on, they turn back off, the engine cranks and starts, the fogs come back on.
Why? The keyless start doesn't just go straight from "off" to on. Like their key-bound fore-bearers, they progress through modes.
When you turn a key, it first goes into accessory mode. This just turns on the radio. For our keyless systems, this occurs when the doors are unlocked/opened or is shut with the fob inside the vehicle. (note: to prevent battery drain this times out after a period and goes back to "off")
Next, it goes into Run mode. From HVAC to engine computers to fog lights, this pretty much activates everything that normally operates while the engine runs. So, a WHOLE bunch-a-clicks. But it only stays there briefly.
After that, it goes into crank mode. Some of the things that were just turned on, get turned off... for example, accessories, fog lamps. Click, click, click, click...
After engine start, it goes back to "Run" mode... where all the things that were turned off, get turned back on. So, clickity, click, clickity, click, click, clickity click click....
So even though it's push button, in many ways it operates like it still has a key.