I mean, it really depends doesn't it?
Using the 73 books of the Catholic Canon, the middle book is Amos, of Amos' 9 chapters chapter 5 is the middle chapter, and of its 27 verses verse 14 is its middle verse. So Amos 5:14 could be argued to be the middle of the Bible.
Even using the Protestant Canon, sans Deuterocanonicals, using the same methodology it's a bit harder since there's an even number of 66 books. Making the "middle book" the 33rd and 34th books, Micah and Nahum respectively here. Perhaps here we could add together the total number of chapters in each book, there are 7 chapters in Micah and 3 in Nahum, which gets us to 10, another even number. So the 5th and 6th chapters of Micah, what are the total number of verses? Well at least now we have an odd number of verses, 31, so the middle verse between Micah chapters 5 and 6 would be (if I haven't thoroughly confused myself yet) Micah 6:1.
But that's just one way of getting to the middle of the Bible. The idea that Psalm 118 is the middle chapter of the [Protestant] Bible is, not really quite correct either. There are 594 chapters up to, and including, Psalm 116, Psalm 117 is the 595th chapter, and beginning (and including) Psalm 118 to the end of the Bible there are another 594 chapters. Making Psalm 117 the middle chapter of the Protestant Bible based on the number of chapters being counted. Since Psalm 117 is divided into two verses, there is therefore no middle-verse.
The history of the Bible has been complicated. We still don't all agree on even how many books actually belong in the Bible as Christians, that alone is kind of a monkey wrench here. And that doesn't even start to address things like how the books should be counted. Are the Minor Prophets twelve books or one book? Are Ezra and Nehemiah one book or two? Or how about verse numbering? How should we number the verses in the Psalms? Depending on the manuscripts we use and, thus the translations we use, how many verses are there anyway (even assuming we could agree on the number of books in the Bible). Do we include, for example, Acts 8:37 or not?
Ultimately, all I'm really saying is that while Psalm 118:8 is a very fine passage of Scripture. The idea of a "Middle of the Bible" is rather malleable, and we certainly shouldn't attach any special significance to it.
-CryptoLutheran