Synonymous hanjas list (most frequent ones)

I think a great start would be using as a reference the certificate which specifically tests for Hanja, the Korean Hanja Proficiency Test.

It has a section that tests for synonyms 동의어(유의어) and another for antonyms 반의어(상대어) and those topics are present from level 6급 (8급 is the lowest of all) to 1급 and special 특급.

There is this blog post which contains some of the synonyms for level 6 (see picture below)

You can see this online lesson for the 한자능력검정시험 which encompasses some of the hanja synonyms of level 3 (which should be already a bit advanced, testing around 2000 characters).

Finally, there is this general list of hanja-based synonym words that appear on multiple levels of the proficiency test.