Well there are lot's of ahadith quoting the existence of a certain hour in Friday in which what you ask in a supplication would be given to you.
There are even some narrations which give a more or less exact timing. And some scholars do qualify some of these timings as better or worse. as you may read in this Fatwa or here in Arabic/Urdu.
But i think supplication is an act between a human and his creator, so if the timing was fixed and well known then why should one do his best to worship Allah and do supplication at any time of the day? I mean somehow it would be strange if people get what they want without an effort. And that's all the same with the hour of the night and the laylatul Qadr, again we know it's a night in Ramadan, and there are ahadith quoting a certain night or certain nights but who really knows which night it is? This is my opinion and Allah knows best!
Here are the most sahih ahadith on the matter:
In Sahih al-Bukhari and Muslim and others all narrated by Abu Huraira.
And a narration by Jabir ibn Abdullah in Sunan an-Nasa'i.
And a narration in Jami' at-Timridhi with an indication for the hour
And a hadith with a different indication in Sahih Muslim
And here some ahadith which have been qualified as hassan from Jami' at-Tirmdihi and Sunan ibn Majah.
And Allah knows best!