"...When the Pontiff has received the tiara, gloves and ring, the archpriest of the basilica, accompanied by two of the canons, presents himself before the Pope, in order to give him a purse of silk embroidered with gold in which there are twenty-five jules of ancient papal money. The archpriest, as he presents the honorarium, says: Beatissime Pater, capitulum et canonici hujus sacrosanctae basilicae, Sanctitatae [sic] vestrae consuetum offerunt presbyterium pro missa bene cantata. Then the hand of the Pope is kissed by the archpriest, and the foot by the two canons. The Pope gives the purse to the cardinal deacon for his train-bearer, who in his turn takes it to the canon sacristan of the basilica, receiving in exchange five ecus, which was about twenty-seven francs before the first World War."
Loosely translated, the text spoken to the pope is: "Most blessed Father, the chapter and canons of this most holy basilica offer Your Holiness the usual offering for a Mass well sung."
Such conveyances for money was not, of course, unique to the popes. In former times, pouches such as these were commonplace in order that people to carry money and the like. The aumônière, or alms-bag, would come to distinguish itself from day to day money satchels, taking exactly the type of form and decorativeness seen here also in the papal aumônière. Here are two historical examples of such:
As was so often the case in traditional papal liturgy then, the papal aumônière is simply a case of the historic conservatism of papal liturgical practice which retained older, historic forms while fads and fashions changed elsewhere in the Church and the world.
So it was then that these special purses continued to be created for the pontiffs, in which to put this symbolic monetary offering. These pouches frequently consisted of embroideries and materials akin to the sacred vestments, coming in either red or white depending on the liturgical colour of the solemn papal Mass in question -- and do bear in mind, these particular liturgical occasions were quite rare.
In our first image, leading this article, we can see a white version of this papal purse, offered by the Franciscans to Pope Leo XIII on the occasion of his jubilee. One will see upon it the year, 1887, as well as the arms of Pope Leo XIII and on the other side, that of the Franciscan order.
Below you can likewise see one of these pouches in red, also from the time of Leo XIII, the obverse side decorated by a simple cross akin to what would be usually found on a maniple or stole.