Abba Chanan said in the name of Rabbi Elazar: "Do not honor a poor man in his quarrel" actually refers to the agricultural gifts owed to the poor — leket (gleanings), shikchah (forgotten sheaves), and peah (corners of the field).
The verse is not about judicial proceedings at all, in this reading. It is about property rights. When there is a doubt about whether a particular sheaf or cluster of grain qualifies as leket, shikchah, or peah — which by law belongs to the poor — you should resolve the doubt in favor of the poor person. These agricultural gifts are the poor person's entitlement, and doubts are decided in his favor.
But "do not honor a poor man in his quarrel" limits this principle. In actual legal quarrels — disputes between the poor person and the landowner — you must not automatically side with the poor. Judicial impartiality is maintained even when the party before you is destitute.
Abba Chanan's reading creates a careful distinction. In matters of agricultural charity, the poor receive the benefit of the doubt. In matters of litigation, they receive equal treatment — nothing more and nothing less. The Torah gives the poor special rights in one domain (charitable gifts) while insisting on strict equality in another (the courtroom). Compassion and justice operate under different rules, and the Mekhilta keeps them carefully separated.