I can't find a 100% definitive source, but according to Darth Bane's Wookieepedia entry, the character's first appearance was a short story titled "Bane of the Sith" by Kevin J. Anderson, which was published in Star Wars Gamer #3. The "cover" published with the story was this full-page image (illustration by Stan Shaw):
I did a quick scan through the PDF version of the short story and didn't find any mention of these black markings around Bane's eyes, so there's still no direct or definitive explanation for why they're there. However, with what knowledge we have available, my best guess would be that they were simply an artistic choice made by Mr. Shaw when depicting the character for his illustration. Since this was the first visual representation of the character, subsequent artists likely took their inspiration and queues from this original portrait to derive their own works.
EDIT: Consider the TVTropes article on Facial Markings mentioned by @NKCampbell in the comments on the OP as a possible - perhaps even likely - motivation for such an artistic choice.
One such artist would be John Jude Palencar who provided the cover art you referenced above for "Darth Bane: Path of Destruction", as well as for Darth Bane: Dynasty of Evil the first and third novels of the "Bane Trilogy", respectively. The cover artist for Darth Bane: Rule of Two, John Van Fleet, also continued this aesthetic for the second book of the Bane Trilogy.
Looking at the original image by Mr. Shaw, I personally wonder if the black was intended to simply be shadows around Bane's eyes - perhaps in the light of the Sith holocron in the foreground - and future artists sort of "ran with it" to define them further as a unique feature of the Dark Lord's appearance.