You won’t find authority for the 10% proposition because it isn’t true. The issue you are raising is how much a single donor can give without “tipping” a public charity into being considered a private foundation.  Here is how it works.

All 501(c)(3) charities are considered private foundations (which can receive all of their contributions from a single donor but are subject to much more stringent regulation) unless they show the IRS that they qualify as public charities under Section 509(a) of the Tax Code.  Sections 509(a)(1) and 509(a)(2) have separate public support tests that consider an organization to be publicly supported if it receives at least one-third of its income from qualified public support. (See Ready Reference Pages: “Calculating Public Support” and “New Schedule A Reflects Change in Public Support Rules”) The tests are run over a rolling five-year period, and the total of gifts from an individual (and certain related persons) over the period are used for the calculation, not the specific gifts from any single year.

Under Section 509(a)(1), if you are not a church, college, hospital or other type of organization that is classified as public because of its activities, you have to receive at least one-third of your total support (excluding related program service income) from qualified sources over the period. Under this calculation a single donor could theoretically give up to two-thirds of total support over the five-year period if all of the other gifts, grants and contributions are fully qualified and equal to one-third of total support.  And since the percentage requirement is not absolute, it might be possible to qualify as publicly supported under the “facts and circumstances” test with more than two-thirds of support from a single donor, but the IRS is not likely to exercise its discretion to do so with so much income coming from a single donor unless the qualifying support is very close to one-third.

Under Section 509(a)(2), the one-third requirement is based on a different formula (which includes related program service income) and is absolute so it is possible that a single donor (who would most likely be considered a substantial contributor whose contributions are not counted at all in the numerator of the fraction) could give close to two-thirds of all revenue, so long as the other revenue is fully qualified for the numerator of the fraction, without tipping the organization to private foundation status. But if the donor goes $1 over the limit, the organization tips.

Some large donors who have been concerned with tipping have made gifts to donor advised funds (DAFs) and advised the public charities that control those DAFs to make gifts to the ultimate recipient charity. Because DAF sponsors are themselves public charities and in legal theory make their own decisions to follow the donor recommendation, those gifts are treated as gifts from a public charity (and not the donor/advisor) and are fully included in the numerator of the calculation. They significantly help the public support percentage.  The IRS is studying right now whether to allow this to continue, however.

While theoretically it may be possible for a single donor to give two-thirds over the period, it is very risky to get very close to that line, in part because the denominator of the calculation includes investment income and other types of revenue that are not contributions. It is a constantly moving target.

By the way, you may want to suggest that your significant donor become a member of an advisory board rather than a director. The donor could be charged with breach of fiduciary duty and found personally liable if he or she never attended a board meeting and things went bad for the organization. A director is required to act in good faith and in the best interests of the organization, and failure ever to attend is a breach of that duty.