The first printed citation for "rush hour" is from 1878 (before the Brooklyn Bridge opened). A popular "rush hour" joke has been cited in print since at least 1957: "Why is it called rush hour when your car barely moves?"
Wikipedia: Rush hour
A rush hour or peak hour is a part of the day during which traffic congestion on roads and crowding on public transport is at its highest. Normally, this happens twice a day—once in the morning and once in the evening, the times during when the most people commute.
The name is sometimes a misnomer, as the peak period often consists of more than one hour and the "rush" refers to volume of traffic, not rate of flow. Typically, rush hour lasts from 6–10 am (06:00–10:00) and from 4–7 pm (16:00–19:00) local time. With people travelling places during their lunch time by car too, it is arguable that noon till 2 pm (14:00) is another, less frantic, rush hour.
The frequency of public transport is usually higher in the rush hour, and in the case of trains, longer ones are often employed. However, the increase in capacity is often less than the number of passengers, due to the limits on available vehicles, staff and, in the case of rail transport, track capacity including platform length. As a result vehicles are more crowded and not everybody has a seat. This may have the effect of making public transport less desirable, therefore pushing more people into cars and making the traffic worse.
Transport demand management, such as road pricing or a congestion charge, is designed to induce people to alter their travel habits so as to minimize congestion.
Similarly public transport fares may be higher; this is usually presented as an off-peak discount for single fares, though season tickets or multi-ride tickets, commonly used in rush hours by commuters, are also sold at a discount.
Merriam-Webster's Online Dictionary
Main Entry: rush hour
: a period of the day when the demands especially of traffic or business are at a peak
(Oxford English Dictionary)
A period of the day during which the movement of people is at its height, esp. one during which large numbers of people are travelling to or from work. Also attrib.
1898 Westm. Gaz. 28 Oct. 8/3 Trailer cars can be put on during the ‘rush hours’, mornings and evenings.
1907 ‘O. HENRY’ Trimmed Lamp 233 As solid as granite in the ‘rush-hour’ tide of humanity, stood the Man from Nome.
1926 Daily Graphic 13 May 1 (caption) The ‘rush hour’ at Earl's Court yesterday. Travelling discomforts are mitigated by much good humour and politeness.
1931 Morn. Post 18 Aug. 6/4 Rush-hour trains held up.
18 December 1878, New York (NY) Herald, "A Remedy Much Needed," pg. 6, col. 4:
Morning and evening there are hours when the rush of passengers to the rapid transit trains overtaxes the proper carrying capacity of the roads under the present arrangements and leads to the dangerous crowding of the cars.
There are several points along the road where central tracks could be laid which would afford facilities for the operation of special trains that would carry off the crowds that collect at some of the way stations during the "rush" hours.
21 January 1882, New York (NY) Times, "The Bob-Tail Car Nuisance," pg. 5, col. 3:
He stated that the company was negotiating with the Stephenson Car Works, and that double cars would be put on the road within four weeks, and would be run regularly during the working or "rush" hours, if not all the time.
5 April 1883, New York (NY) Sun, pg. 1, col. 5:
"Mr. Wiman announced a desire to make all the fishing grounds surrounding the island, including prince's bay, Raritan Bay, the lower and upper New York Bay, as accessible to New Yorkers as Coney Island is. More frequent trips, commutation rates, and more boats in rush hours are promised."
18 April 1883, New York (NY) Sun, "Some Staten Islanders Disappointed," pg. 1, col. 6:
The new management of the Staten Island Railway Ferry Company announced yesterday that hereafter four extra trips would be made by the boats during the rush hours of the morning and evening.
13 January 1885, New York (NY) Sun, pg. 2, col. 6:
Superintendent Martin told the trustees that during the rush hours the trains were run on 1 3/4 minutes headway. This was as fast as they could be run with safety. The cars were absolutely packed in the mornings and evenings.
6 June 1886, New York Times, pg. 2:
The conductors and drivers who handled them were all union men, who said that they had been ordered to take a few cars out during the rush hours if the company desired.
26 September 1886, New York (NY) Tribune, "The Long-Suffering Public: Inconvenience of reaching the Brooklyn Elevated Railway from the Bridge," pg. 12, col. 2:
During rush hours the south landing is almost constantly packed. Supposing one-third of the people, a fair estimate, wanted to take the railroad.
20 July 1887, Cedar Rapids (IA) Evening Gazette, pg. 2, col. 3:
One evening during the "rush hour" on the Brooklyn bridge a young woman, neatly and tastefully dressed, succeeded by some strange oversight of the men in securing a seat in one of the cars.
(From the New York Tribune -- ed.)