The following is just one of the examples of thinks I being used by Melville:
When, at last, his mind seemed made up touching the character of his bedfellow, and he became, as it were, reconciled to the fact; he jumped out upon the floor, and by certain signs and sounds gave me to understand that, if it pleased me, he would dress first and then leave me to dress afterwards, leaving the whole apartment to myself. Thinks I, Queequeg, under the circumstances, this is a very civilized overture; but, the truth is, these savages have an innate sense of delicacy, say what you will; it is marvellous how essentially polite they are.
Thinks I occurs thirteen times, whereas thought I occurs thirty seven times in the novel. Methinks the phrase is used by Melville in the meaning of thought I. What puts me in doubt, though, is the lack of definitions of such meaning in dictionaries and the lack of any discussion in the Web, e.g. at ESE. I intend to edit Wiktionary’s entry on thinks as soon as I learn from you all..
Thinks I is an example of the historic present tense. This tense is used to add immediacy to the narrative. https://en.wikipedia.org/wiki/Historical_present.
Methinks/methought seems to have been passing out of the language at the time that Melville wrote, and "Thinks/Thought I" seems to be a transitional form.
The third person form of the verb seems to be a colloquialism based on the (lack of) conjugation of "methinks" (impersonal) = "it occurs to me."
The commoner form "says he" presents no such problem.