The gemara[1] says: “[If one declares]: ‘This coin shall go to tzedaka’ – before it reaches the gabbai’s[2] hand, it is permitted to modify it; once it reaches the gabbai’s hand, it is forbidden to modify it.” The gemara[3] challenges this from the story of Rabbi Yannai, who borrowed tzedaka money after it had reached the gabbai. It answers that he did so in order to indicate to others that there were no liquid funds, so that people would be encouraged to give more. Thus, his borrowing helped the poor. Rishonim argue about the meaning of “modifying” money. Rashi and Tosafot[4] explain that the donor could lend the money to himself or to others before he gave the assigned money to the gabbai. The Rambam[5] seems to relate the gemara to switching the coins to different denominations but with the same cumulative value. The Beit Yosef[6] has difficulty fitting the Rambam’s explanation into the gemara and incorporates only Rashi’s view into his halachic work.[7]
There is much discussion among Acharonim about the question whether the status of a tzedaka box is like that of a gabbai. Without delving deeply into this question, we mention that most treat it like a gabbai.[8] Therefore, it might seem that our question depends on the differing approaches to the aforementioned gemara. According to the Rambam, one may not even switch the coins in a tzedaka box. According to Rashi, which the Shulchan Aruch accepts as halacha, whereas apparently one may not borrow the money, he is permitted to simply make change. After all, tzedaka money does not have intrinsic holiness that would make it religiously forbidden to use.[9] On the other hand, it is likely that Rashi disagrees with the Rambam only regarding the interpretation of the gemara, not the halacha,[10] and after the gabbai receives the money, making change from tzedaka funds is like doing so with a friend’s money to which you have access, without his permission.
So then, is one permitted to use money that someone entrusted to him? The Shulchan Aruch[11] states that whether or not he can borrow the money depends on whether the watchman is one who deals in coins regularly and on whether the owner hinted about his feelings on the matter. However, as a rule, he may not borrow it. He does not explicitly address the question of exchanging coins, which Acharonim dispute.[12] Therefore, regarding our case, it is unclear whether one may use the cash in a tzedaka box to make change even when we know of no specific reason to suspect that the intended recipients will thereby lose. This justifies the stringent practice you cited. When leaving even a small donation, the act is considered giving tzedaka rather than taking change, and it is permitted.
Besides the fact that not all agree to the stated reasons for stringency, additional factors play a role. Concerning a general tzedaka box that the shul’s gabbai administers at his discretion, there may be an understanding that the money can be used for such things as getting change.[13] This may depend on the local practice. We should also recall the gemara’s idea that if the poor can gain from the money’s use, it is permitted. The question is whether by using the tzedaka box to get several coins in change, of which some will be distributed to the poor, one will, overall, give more tzedaka. One must also factor in whether enough coins remain to enable the next person to give a donation that requires change.
In summary, we have seen both the logic behind the stringent practice you reported and possible grounds for leniency, especially under certain circumstances.