In 2010, you came across a site that was giving away bitcoins. The site said it was free money so you signed up. You were logging in to figure out how it was money but couldn't. The last time you logged in you had more than 100 bitcoins. Later you forgot about the site until 2017 when a ponzi was advertising Bitcoin. Bitcoin? The name rang a bell and you stated hearing stories. You remembered you must have many bitcoins somewhere but you can't even remember the name of the site. How would you recover your BTC?
Name of the website doesn't matter in any way. According to what you said (in imaginary) you had bitcoins on website's account which are just numbers before withdraw.
All you can do is search for websites and if you really search, you may find url and then put that url on web.archive.org and the best you can do is to find website's contact page (email of owner) but more likely it will be email with domain's name and even your attempt to contact owner will fail. Even if you contact, what do you think, will he give a f*** to what you say?
To make it easier for you, you can't make person from dead to alive, so the same applies to your bitcoins.
And even if you could somehow contact the owner of that website that used to store your coins there is no guarantee either that they are still holding those coins and even if they are what makes anyone think that they would be very happy to give away 100 bitcoins now that they are so valuable? While the scenario is not real and it is just an imaginary one it is important for one reason, you need to be the one that store your coins if you do not then you are relying on someone else to do it for you and in this market that is a mistake.
Bitcoin was supposed to give us freedom but with freedom comes responsibility and you cannot let anyone store your bitcoins for long and expect to not lose those coins.