What I do is to click on the "like" or "favorite" icon.
But, I have close to 4,000 likes on Twitter and it is almost impossible to find older likes using the regular Twitter site.
I have several different solutions to this problem.
Using TweetDeck I created a column for my likes. TweetDeck columns have the option to search just that column for any content I am looking for.
In the example below I am searching all 4,000 of my likes for anything that has the hashtag #GoogleClassroom
I can also search for all the Tweets I have liked from specific people. In this case I looked for all the Tweets from @jmattmiller that I have liked.
Another way I manage all the Tweets that I have liked is to use IFTTT (If This Then That) - http://ifttt.com
I have created several IFTTT Applets that do specific tasks whenever I "like" a Tweet.
These Applets will take the Tweet that I have liked and a) append the tweet to an Evernote note b) add the Tweet to a new line in a Google Sheet, c) append the tweet to a Google Doc in my Google Drive and d) add a new bookmark to my Diigo account
This way I can easily copy and past Tweets that I want to share with other. I often copy them into shared documents based on interest for math teachers, language arts teachers, GSFE users, STEM, etc.
If you haven't used IFTTT before, it is extremely easy to create your own Applets. Basically you just pick and chose what you want to have done.