A vocal minority of Christians claimed that the world would end Oct. 7, 2015. What was the basis for their belief? What passages or reasoning were they using to support this claim?
This appears to a revised prediction based on an earlier failed prediction from Harold Camping. Camping predicted that the "rapture" would occur on May 21st 2011, followed by five months of calamitous events before the world would end on October 21st 2011. After this failed to eventuate, He kept a relatively low profile until his death in 2013; however followers of his system (in particular, Chris McCann of the eBible fellowship) modified it slightly, claiming that the end of the world would be 1,600 days after the "beginning of Judment Day" on May 21st 2011, bringing the revised date of the end of the world to October 7th 2015.
One of the bases of the system that Camping employed to come up with his dates was from a particular day-age interpretation of Genesis 7:4 -
Seven days from now I will send rain on the earth for forty days and forty nights, and I will wipe from the face of the earth every living creature I have made. - Genesis 7:4 NIV (emphasis added)
He interpreted this to have a double fulfillment, meaning that as well as Judgment coming (by the great flood of Noah's time) in seven days, there would also be a final Judgment that would come after 7,000 years (applying a literal interpretation from Psalm 90:4/2 Peter 3:8 that to the Lord, one day is as a thousand years).
His system of fixing both the beginning of the world at 11,013 BC and the date of the flood at 4,990 BC diverged from Ussher's dates (4004 BC / 2349 BC) and was based on a radical reinterpration of the patriarchal genealogies in Genesis 5 & 11, interpreting that the underlying original language implied that (some) of the descendants were not direct (ie sons) but that the lists skip generations, with some of the lifespans starting from the death of the preceding "reference" patriarch rather than much earlier in their lives. I admit that I don't understand his logic at this point myself, a few more details are available at the linked (Wikipedia) article, but if you want more of the story, you would have to either read his book Adam when? or the article(s) linked in the question.
From there, adding 7,000 years gets us to the ballpark (depending on year length chosen and type of calendar used etc.) of the present where Camping identified 1988 as the 13,000th year of human history and an important milestone with regard to calculating the final timeline of human history, being the end of "the Church Age" - it certainly was for him, as this was the year he left the Christian Reformed Church, later declaring:
that all churches have become apostate and thus must be abandoned. He encourages personal Bible study and listening to his Family Radio broadcasts - https://en.wikipedia.org/wiki/Harold_Camping
He made his first prediciton for the return of Christ as being on September 6th 1994. After that date came and went, Camping revised it's significance to being the start of the great tribulation and eventually came up with the dates mentioned in the first paragraph.
After admitting to being flabbergasted that things hadn't turned out the way he'd expected, Camping apologized in March 2012 - "We humbly acknowledge we were wrong about the timing." and announced he would no longer make doomsday predictions. However after his death, his family and supporters vowed to continue to promote his teachings.