The BHS of Jeremiah was edited by Wilhelm Rudolph, a distinguished Alttestamentler, and the author of several important commentaries -- among them, a commentary on Jeremiah. It was first published in 1958, with the last edition being the third which appeared in 1968. His BHS edition of Jeremiah was published in 1970.
As OP notes, Rudolph suggests hayyôm ["the day"] for hāʾîš ["the man"] simply as a "proposal" -- a "conjectural emendation", as these are sometimes called. There is no manuscript or versional evidence for it (unlike his note on the preceding word which is widely recognized and adopted).
The notes in BHS did not allow for providing rationale for "prp" notes. This is where Rudolph's commentary comes into play. There he offers a fairly full explanation (p. 132), giving "'Tag'" ("day") as his translation, he comments:
𝔐 "Mann", aber es ist sehr auffallend, daß gegen den Mann von v. 15, der doch nur eine Nebenfigure ist, eine solche Drohung geschleudert und ihm (17) der Vorwurf gemacht wird, daß er das Kind nicht noch im Mutterleib getötet habe: wie konnte er das? und waren bei Entbindungen fremde Männer anwesend? Die von den meisten Neueren vorgezogene Korrektur הַיּוֹם erleichtert das Verständis: der Geburtstag wird wie ein persönlicher Feind vorgestellt.
Rough translation =
[The MT has] "Man," but it is quite striking that a threat like this is cast against the man of v. 15, who is only a secondary figure, and that against him the accusation made (17) of not having killed the child while still in the womb: how could he do that? and were there any foreign men present at the delivery? The correction הַיּוֹם, advanced by most of the recent [commentators (perhaps Rudolph has Artur Weiser in mind?)], clarifies the sense: the day of birth is presented as a personal enemy.
To this is appended a lengthy footnote interacting with an article by Leo Prijs which isn't necessary for this Q&A.
Rudolph's suggestion has been challenged, and at length by William McKane, who devotes several paragraphs to this as part of the complex of textual problems in this verse and its immediate context (A Critical and Exegetical Commentary on Jeremiah. Vol. 1 Jeremiah I-XXV [ICC; Edinburgh: T & T Clark, 1986], pp. 484, 487-8).
McKane concedes that smoothing out the metaphors in vv. 14-18 has some appeal, but that the emendation is not warranted: the "day" metaphor is submerged by martial imagery, spotting an allusion to Sodom and Gomorrah in the "did not pity" phrase. McKane's final judgments are sometimes difficult to discern, but he summarizes this way (p. 488):
Is this representation [he's referring to his sense that the passage connects to Sodom and Gomorrah] better served by היום than it is by האיש? If האיש is read the sense is: May the man who brought news of the birth be like Sodom and Gomorrah which were overthrown by Yahweh's judgement on a day when they rang with cries of distress issuing from their citizens. If היום is read, we have to, [sic] assume that the simile is imperfectly articulated and that what is intended is something like: May that day be like (the day on) which Yahweh overthrew Sodom and Gomorrah; may it hear cries of distress in the morning, shouts of anguish at noon. In view of the fact that the versions support האיש, that there is no other textual evidence in support of היום, and that האיש is present in v. 15 in any case, an emendation of MT is unjustifiable.
So at least in this case, McKane's judgment is clear -- and er hat recht. ;)