I asked my accountant this question after I joined the Acorn cult
He told me:
"Since you pay no tax on the donation, you would usually want to donate your lowest-basis shares which have a long-term gain. If you donate non-covered mutual fund shares, Vanguard will only provide average-cost information. If you are using specific identification of shares for tax purposes, you need to identify which shares are being donated in the same way; send Vanguard a secure E-mail, or enclose a letter with the form, which says, "Please donate $10,000 (or 123.456 shares) from the shares purchased on 1/2/04." If you donate covered mutual fund shares, Vanguard will use your preferred lot identification method; if this is specific identification, you will have to include a similar letter because there is nowhere to select lots on the form. If you donate securities from your brokerage account, you can specify lots on the form itself, and you can use specific identification even if this is not your default method. (For example, you might want to sell shares automatically as highest-in-first-out, which usually minimizes taxes; when you donate, you would specify your lowest-basis shares.)"
So I have a list of my past purchases, ie shares purchased on so-and-so date. I list those shares on the form when I submit to the Church DIK office. It is a faxed form each time. I do this once a year, usually early in the year to "stay ahead" on tithing, but as Acorn clearly states we can try to time it so that we harvest gains as much as possible.