What method(s) of reasoning did Abraham use to discover HaShem's existence and that he is Adon Olam? I'm wondering about particular methods.
the shaar yichud of chovos halevavos has a systematic logical inquiry into this subject. the manoach halevavos commentary says in ch.10 "the truly wise have a single, common viewpoint, and all of them grasp exactly the same matter with only these few hints, according to what is fitting and possible for them, because the false and erroneous ways are numerous but there is only one way of truth."
the shaar yichud has multiple approaches from logic and also from the wisdom and interconnection of nature. they all lead to same conclusion. Avraham probably also considered all evidence including wisdom in nature, inquiry into the Unity, etc.
From a Torah perspective, he didn't need to discover the answer, he just needed to un-learn the un-answer given by his parents et al.
According to Rav Hutner, he did this by extrapolating the middas ha'chesed he found within himself to its source.
I.e. he discovered ha'adam nivra be'tzelem, and he found the nimshal of the mashal of his own life.