There’s actually a “bookmark this tweet” option under the more options menu for the tweet.

Bizarrely, this appears to be missing from the main site...

Ah, I basically only use the Android app, and this is hidden under the button that one expects would launch the confusing and awful Android sharing dialog.

This is kinda useless. You can bookmark a tweet. But when you undo this bookmarking, all other bookmarked tweets also go.

I don't have this option on tweetdeck, only a add to collections, gives me a popup with no content. Nice.

It’s on the official site, I’m not sure of 3rd party app support. To view your bookmarks, https://mobile.twitter.com/i/bookmarks

I only see it on the mobile site.

It's found to the right of the heart button.

tweetdeck is technically first party, twitter acquired the team many many years ago

I recommend to use a real bookmarking service like Kozmos (https://getkozmos.com) or Pocket instead of Twitter’s bookmarking tool. You can’t search and organize anything on Twitter, and you’ll lose them when you delete your account like I did.

P.s: I’m the founder of Kozmos, but honestly would promote any bookmarking service over Twitter features.