A calque of German ‘blauer Montag’, ‘blue Monday’ originally denoted a Monday on which people chose not to work as a result of excessive indulgence over the course of the weekend. Under the influence of the adjective ‘blue’ in the sense ‘dismal’, it came to denote a Monday that is depressing or trying.
—Cf. also Mondayitis.

 

The name blue Monday originally denoted a Monday on which people chose not to work, especially for a celebration or as a result of excessive indulgence over the course of the weekend; a calque of German blauer Montag, it is first recorded in An Historical Development of the Present Political Constitution of the Germanic Empire (London, 1790), the translation by Josiah Dornford (1762/3-1797) of Historische Entwickelung der heutigen Staatsverfassung des Deutschen Reichs (Gottingen, 1786-87), by Johann Stephan Pütter (1725-1807):

It was formerly, and in many countries it is still the custom in Germany, for the journeymen, &c. employed in the lower kinds of trade, to consider every Monday as a day set apart for idleness, and no inducement can prevail upon them to apply themselves to work. Perhaps the custom was derived from the post-festum mentioned in the Canon Law; and the expression “Blue Monday” is supposed by some to have its origin in the bruises occasioned by the fist and cudgels, which were in frequent use among the drunken and disorderly; but, as we meet with a Blue Tuesday likewise, the derivation seems more probable which occurs in a manuscript Thuringian Chronicle.
“In the sixteenth century, it was the custom in Germany to ornament the churches on fast-days with blue; and at this period the tradesmen began to keep the fasts by neglecting their work. This was not only usual among the master tradesmen, but they indulged their servants likewise in the same privilege. For want of employment, the common people had recourse to drinking; and, instead of fasting, it soon became a common proverb, Heute Blauer Frassmontag, To-day is feasting Monday.”

This text, therefore, connects the German name blauer Montag with blue as a liturgical colour associated with penance in the Lutheran and formerly also the Catholic church. However, according to the Oxford English Dictionary (3^rd edition, 2013), this interpretation seems less likely, for chronological reasons, than a reference to blue clothing worn on Sundays and holidays—a reference supported by expressions such as Middle High German des vīretages blā, literally holiday blue, with allusion to clothing.

The earliest use of blue Monday in a non-German context and in the same sense that I have found is from The Long-Island Star (Brooklyn, New York) of 1^st March 1827; “an old apprentice” gave several recommendations about New York apprentices, among which this one:

Let all workmen have their wages paid to them on Monday, instead of Saturday night. The possession of a week’s wages is an incentive to extravagance and dissipation which few of them can withstand on a Sabbath; if paid in the beginning of the week, it will generally be better applied. This plan is now successfully pursued in some of the best regulated workshops of our city: and the consequence is, that since the attempt was first made, blue-Monday has not been observed by a single man, in at least one establishment.

From the fact that people chose not to work, blue Monday came to designate Monday, as characterised by its low level of activity; the earliest occurrence of this sense that I have found is from The Weekly Courier and Journal (Natchez, Mississippi) of 21^st April 1837, which explained that, when the New York Stock Exchange had opened on Monday 3^rd April:

Stocks, it will be seen, have fallen off a little to-day.—Blue Monday, as it is commonly called, is apt to be dull in stocks.

The following from The Sun (Baltimore, Maryland) of 19^th June 1838 shows the correlation between low level of activity and high level of absenteeism, as well as the existence of the phrase to keep blue Monday:

                                                                                                                           Monday, the 18^th.
There was but little business transacted in the Court to-day, the witnesses and parties litigant seemingly having made up their minds to keep blue Monday, and keep as cool as mint juleps and ale sangarees could render them.

This advertisement was published in the Public Ledger (Philadelphia, Pennsylvania) on 18^th January 1851:

Cigar makers wanted—On good work. Those who keep blue Monday need not apply; fair prices given, at 154 North Third St.

The name blue Monday also came to designate Monday, as the day on which one suffers from the effects of overindulgence in alcohol; for instance, on 30^th January 1846, the Daily National Pilot (Buffalo, New York) published a correspondence from Washington, dated 24^th January, which contained:

To-day, though Saturday, has been a blue Monday;—headaches have been quite prevalent.—Strong drinks have been eschewed in favor of mild, alterative Seidlitz, or corrective Soda.

This led, probably under the influence of the sense dismal of the adjective blue, to the current meaning of blue Monday, that is, a Monday that is depressing or trying. The earliest occurrence that I have found is from The Sun (Baltimore, Maryland) of 20^th November 1849; on the previous day, Monday 19^th, its New York correspondent had written:

We are enveloped in dull gray blanket-like clouds, which are slowly distilling their watery contents upon all Gotham, rendering this the commencement of a truly blue Monday. The first snow storm of the season may soon be expected if the clerk of the season does not give us a fresh instalment of Indian summer.