I suppose it depends on what exactly you're looking for. "The History of [X]" is almost always going to be a very broad category, and can range from the rather dry and granular (here's the date the UN was founded, here are its founding members, here are the major challenges it has faced in the years since,) to the broader and more speculative (here's a history of extra-national authority, here's why the League of Nations failed, here's the broader political context of the world in which the UN came to be). I don't have a recommendation for the former, but I might have one for the latter.
While not explicitly (nor exclusively) about the United Nations in and of itself, you might get some use out of Strobe Talbott's 2008 The Great Experiment: The Story of Ancient Empires, Modern States, and the Quest for a Global Nation.

In it, Talbott (former deputy US secretary of state and then-head of the Brookings Institute) essentially examines the evolution and concept of extra-national authority, beginning in pre-history with the 12 tribes of Israel, and eventually progressing through to the modern era (or at least modern at the time of writing, which was the GW Bush Presidency, though I found his predictions for the near future to be surprisingly on point) with the larger context of 21st century globalization and the challenges posed to it by American unilateralism.

While only about a quarter of the book is dedicated solely to the UN (and I will warn, the book is more than a little Amero-centric, both due to Talbott's own perspective, as well as the simple fact of the USA's status as a "unipolar" superpower post-Cold-War,) it does give a good overview not only of the body's founding, and the challenges surrounding it, but also both its usefulness and limitations as a modern extra-national governing body (the dust-jacket quote refers to the UN explicitly as "flawed but indispensable").

In short, if you're looking for a how book exclusively on the history of the UN (names, dates, timetables, etc.) you might be better off looking elsewhere, or perhaps checking out The Great Experiment from the library and just skimming the later chapters. But for a much greater contextualization of the why of the United Nations, (the growth of tribes into nations into empires, the failure of the League of Nations, the greater issues and attitudes surrounding globalism, the balancing act of national vs. extra-national sovereignty, and overall the broader historical context of international power in which the UN exists,) I really don't think you can do better.