Thank you for your reply. I arleady solved this problem. Here is the solution (it will maybe help others who are experiencing similar problems): Ad Blocker in my browser was causing this situation - the problem was solved when I turned it off (so, it turned out that when the Ad Blocker is on, the blue "Select" button on Swisscom website does not work. Strange...)
I have the "inOne Home" Internet subcription, and in paperwork that I received from Swisscom I can see that I can use STANDARD MyCloud free subscription (250 GB). I made new login account this evening in order to start using MyCloud for the first time (with the same email address I use for Swisscom). When I login in MyCloud, I can see that LIGHT MyCloud subscription is set for me. But, when I click on "Change subscription", and on next screen when I click on the blue "Select" button below STANDARD Subcription, nothing happens (as if the "Select" button is not working). I am using latest version of Firefox browser. Any ideas what could be the problem?