I often see donation requests with amounts in multiples of $18. I would like to know the source for giving charity of the numeric value of the Hebrew word chai (18) and its multiples.
When charity is given as a form of merit, as opposed to a general tithing obligation, it is seen as an intercessor before G-d. This is termed a sanegor.
Angels also play a part in this role, especially the angel, Michoel. To attest to this symbolically or perhaps to request that this angel intercede and serve as a sanegor on our behalf there is a custom to give charity with the numerical equivalent of Michoel - 101.
This follows the allegory told by Hillel to Bar Hei Hei of the wagon driver who charges his fare a specified fee for driving 10 parasangs, but doubles the fee for driving 11. Because 1 parasang beyond his habit goes completely beyond his nature, it has double the value of acting only according to ones habitual nature.
It is worth noting that this concept is also associated to 101 as found in regard to the difference in reviewing what one has learned in Torah 100 times versus 101 times like is found in Chagigah 9b. Hillel explains that the individual who goes beyond their natural habit (for example, their natural habit of giving, like the required tithe) is one who serves G-d.
So the idea of 101 is connected to the doubling of payment or blessing. This custom is still prevalent among Sefardic Jews.
Since giving 101 of a particular currency may not be feasible because of excessive cost to the individual, other numeric equivalents which were connected to angels were found.
18 stems from a kabbalistic understanding of a story in Chagigah 15b: Rebbi Mayer found a pomegranate (רמון) he ate the inside and cast away the husk.
The kabbalists understood this to be a veiled reference to the angel מטטרון שר הפנים [MaTat, the leader of the entire Angelic court as alluded to by the leading letters to his full title which spell Moshe, משה, who is the leader of the Jewish people]. Just as the Shechinah (G-d's revealed presence in the lowers worlds) spoke from the midst of Moshe's throat, so to the inner aspect of Metatron is equated with the Shechinah in his realm.
The idea alluded to with the story of Rabbi Mayer is that the letters of the word רמון are the outer letters of the name מטטרון. When those are removed you are left with ט-ט, the inner substance or פנימיות, which is what Rebbi Mayer "consumed". In other words, the inner essence of the Angel Metatron is the Shechinah. This is alluded to by the gematria value of ט-ט which is 18 or life.
Hence, the giving of charity in multiples of 18 (doubling 18 for example like with the Angel, Michoel). And this is derived from the the posuk from Bemidbar 14:28 which says:
אֱמֹ֣ר אֲלֵהֶ֗ם חַי־אָ֙נִי֙ נְאֻם־יְהֹוָ֔ה אִם־לֹ֕א כַּאֲשֶׁ֥ר דִּבַּרְתֶּ֖ם בְּאׇזְנָ֑י כֵּ֖ן אֶֽעֱשֶׂ֥ה לָכֶֽם׃
I am life, says G-d...
While not directly answering this question, there is an incident recorded in R. Hayyim Joseph David Azulai's diary that shows that the gematria of the sum of charity was taken to be significant. Once you have that premise, it is not a huge leap to specifically the value of chai, as I think most people would readily acknowledge "life" to be the the most fundamental blessing. Multiples of chai is surely just a way of amplifying the message of "life", (as opposed to an entirely separate rationale). Here is the text of the diary entry for November 21st, 1753:
'And the reply came back' from the eldest among them that they had reached agreement, 'and a spirit of goodwill supported them' for the sum of two hundred and seventy five Corinthian ducats [= escudos]. I answered that 'I went through the sum' mentioned, and 'there is a hint of the number' of evil, and it is not fitting to repay evil for so much good – 'for they had rendered me only good'. And when they heard me present this pleasantry it found favour in their eyes, and they added another five! So I left there with 'behold! it is good.
In his work, The Book of Ruth: Paradise Gained and Lost Benjamin Segal makes the observation that in the book of Ruth God's name appears 18 times (as it does in Tehillim 29)
I would like to suggest, based on Chazal who say that the book of Ruth teaches the importance of chessed, perhaps the notion of donating 18 or a multiple thereof is related to this