The method credited to Ibn Abbas is philological before it is theological. Asked about a rare or difficult word, he is reported to have answered by citing a line of pre-Islamic Arabic verse in which the same word appears, on the principle that poetry is the register of the Arabs and preserves the vocabulary the revelation was delivered in. A celebrated set of two hundred questions on obscure words, put to him by a Kharijite questioner and answered from poetry in each case, circulates under his name as an illustration of the method whatever its own transmission history.
Two other strands are associated with him. One is the occasions of revelation, the reports of the circumstance in which a passage was given, which anchor a verse to a moment and constrain how widely it can be read. The other is the identification of what a pronoun or an allusion refers to, which is the bulk of early commentary.
The discipline that grew out of this had to spend centuries sorting his name from the material attached to it. Later critics of tafsir graded the chains of transmission from him explicitly, treating one written collection transmitted through Ali ibn Abi Talha as the soundest available and rejecting another route outright as a chain of falsehood. A large commentary circulating under his name is generally judged by scholars not to be his work.
That sorting problem is itself the point of interest. The most cited authority in Quranic exegesis is also the authority in whose name the largest quantity of doubtful material was produced, and the two facts are connected.