In short, in 1891 the new federal Board on Geographic Names sought to impose consistency with similar names, such as Harrisburg. The general rule is on p. 20 of this document. But Pittsburghers resisted, complaining that the board’s decision was based on faulty sources, and that changing the Scottish burgh to a German burg was rewriting the history of the place and its founders. The city, the state, and many local institutions continued to use the h. Pressure from Senator George T. Oliver and others eventually persuaded the board to reverse itself, and the Pittsburgh spelling became official on July 19, 1911. Here’s toponymic scholar Mark Monmonier’s brief recounting.

The Wikipedia entry Name of Pittsburgh tells the story in substantial detail, with good citations to the relevant sources. Many, like the decisions of the Board on Geographic Names, are easily accessed online. If you want to know more about the history of the Board, here's a scanned PDF of a 1990 symposium marking its centenary.

There's a section on this topic in George R. Stewart's Name on the Land (Chapter XXXVIII Of Rules and Regulations).

First, you wrote 1880, but it was really in the 1890s. Up until 1890 there was little effort toward standardizing place names. Places, especially natural features, might have several commonly used names and a wide variety of spellings. The Post Office provided some high-level structure for the naming of towns. One of the first things the founders of a new town would do is open a post office, which involved submitting the town name to the Post Office. If the Post Office thought the name would be confusing for them they might deny it, perhaps offering an alternative name. This might be done if another post office already had that name in the region, or for some other reason the Post Office thought the proposed name would make it hard to properly deliver mail. But the function of the Post Office is to deliver mail and they try their best to do so regardless of how the address is written or spelled. Multiple names and spellings still proliferated. Further, a town might continue to use its "original" name despite the Post Office having given it a different name. There were many disputed names.

In 1889 a group of representatives from a number of government agencies involved with making maps met to discuss problems relating to place names. These agencies included the Geologic Survey (USGS), the Coast and Geodetic Survey, the Hydrographic Office, the Lighthouse Board, the State Department and War Department, the Smithsonian, the Post Office, etc. It had become an increasing concern how a map made by one of these agencies might bear different names from other maps of the same place made by other agencies. The group decided that a centralized place name authority might be the best solution. And so in 1890 President Harrison established the Board on Geographic Names (BGN). The Board had no power to enforce its decisions on the population at large, but its decisions were compulsory on all government offices, which generally gave the decisions a sense of authority and were almost always followed by people in general.

The main duty of the BGN was to settle name and spelling disputes. For example, a mountain near San Diego was called both Cuyamaca and Cloud Peak. The BGN decided in favor of Cuyamaca. In South Carolina there was an island spelled Daufuskie or Dawfuskie. Not a big deal but still an "irritation", especially when the USGS used one spelling and the Post Office another. In its first year the BGN made about 2,000 decisions of these types, and for the majority of them it is hard to see any problem. They were solving a real problem.

Very quickly the BGN found that it had to state some standards. There were so many names to be dealt with they couldn't really make decisions one by one without any guidelines. So in 1891 they presented 13 principles for place names in the United States. The first was that local usage in spelling and pronunciation should be used as much as possible. Others dealt with disputes. Where local usage was split preference should go to the name that was "most appropriate and euphonious". Nine of the principles were specific recommendations, which included: Names should avoid possessive forms (Pikes Peak instead of Pike's Peak), diacritic characters and hyphens should be avoided, names of two or more words should be written as one word, "Centre" should be spelled "Center", and names should drop "City" and "-town", the "-ugh" of "borough", the "-h" of "-burgh", and a few other things.

These last guidelines were intended to create uniformity and efficiency, but they caused the most difficulty. While they generally reflected the natural evolution of naming that had been taking place anyway, they often undermined that very evolution. As Stewart puts it:

Vermonter or Missourian alike might not care a penful of ink between -borough or -boro. In fact, as a practical man, he might even prefer -boro. But if anyone told him he must use -boro, he was likely to invoke all the shades of the Founding Fathers in asserting his inalienable right to -borough.

Still, for a while the BGN proceeded quite successfully in implementing these kind of changes. And they did not fight against well-established local usage even when the same name might be spelled differently in different places. Like the spelling "Alleghany" in Virginia, "Allegany" in New York, and "Allegheny" in Pennsylvania. In general they favored local usage over the nine "guidelines" of standardized spelling.

In 1894 the BGN began to apply the "nine guidelines" more vigorously. They began a wholesale standardization of post office names according to their spelling guidelines. Mostly their decisions affected new post offices and were accepted without question. But when they tried applying respelling to older, established towns, things were not so easy. And one of the most problematic changes was making -burgh into -burg. In most cases, -burgh was the Scottish spelling and -burg German. Stewart says this change "seemed to Germanize the country overnight." Some towns accepted it without complaint, but some resisted. For example, in the official post office list, Newburgh, NY, became Newburg in the 1895 list, but was back to Newburgh in 1896. Local sentiment had been strong about the spelling.

Then Stewart goes on to talk about "the largest -burgh to be affected", Pittsburgh. He points out that although it was from early times a stronghold of Scots, "Scots-Irish", and the Presbyterian Church, its people had never really seemed to care whether it was spelled Pittsburgh or Pittsburg. An 1816 charter spelled it Pittsburg. In 1890 some newspapers used the -h while others did not. The Post Office used Pittsburgh. In short, local usage was split and there was no obvious historical standard to fall back on. Further, the Presbyterian Irish in Pittsburgh were not, Stewart says, as conscious as some Irish and Scots elsewhere "that the -h was part of their birthright".

Stewart says that if left to itself it would probably soon have worked out a solution. But the BGN rendered a decision for "Pittsburg" in, I believe 1891. However, the people of Pittsburgh were "a notoriously stiff-necked and democratic people; if told by self-styled higher authority to do anything, they are very likely to do the opposite." For several years there was confusion and people began to "take sides". The BGN officially changed the post office name to Pittsburg in 1895, but they had no power over the Pittsburgh Gazette, the Pittsburgh Stock Exchange, or the University of Pittsburgh. The Post Office "fought back" by pointing to a whole string other other towns from New Hampshire to California that had consented to "Pittsburg".

As the years went by and the question continued to cause problems and arguments, an official investigation was done into historical sources, which concluded that the -h form had tradition behind it. Finally in the end, Senator George Oliver took up the case, advocating for "Pittsburgh". A group of "many prominent citizens" supported and pressured the senator.

On May 27, 1911, Chairman Gannett of the BGN wrote "Senator Oliver is very anxious [...] to have the name changed." Then on July 19, 1911, the BGN officially reversed its decision and declared for "Pittsburgh". The Post Office followed. And that's basically it.

Stewart goes on to say that although many hundreds of post office names were changed, the attempt at standardization was a "flat failure". It was most successful in the elimination of the possessive form, which most Americans were omitting by that time anyway. But even here there are some cases where the possessive was retained. The Board's current FAQ says that since 1890 only five possessive names have been allowed by Board decision: Martha's Vineyard (1933, "after an extensive local campaign"), Ike's Point, NJ ("because it would be unrecognizable otherwise), John E's Pond, RI, and Carlos Elmer's Joshua View (to avoid confusing the tree Joshua with the name). Apostrophes were retained in things like L'Anse, O'Brien, and Point O' Woods. Diacritic marks were largely done away with, yet there are examples of places like Cañon City, which refused to become Canyon City and is today Canon City.

Probably the least effective of the "guidelines" was combining multiple words into one. Today there are about an equal number of Newcastles and Lafayettes as New Castles and La Fayettes. Many places submitted at first but over time reverted. For many years the post office list had a constant return to two word place names. As Stewart puts it:

The instinct is sound. Glenlyon and Deeprun are distinctly something else when run together. As a single word, Bigprairie is grotesque. Coscob is simply not Cos Cob. Even the [BGN] did not attempt Newyork, or Rhodeisland.