Here is an Ngram chart for the expressions "advantage of doing" (blue line) versus "advantage in doing" (red line) versus "advantage to doing" (green line) versus (just for the heck of it) "advantage from doing" (yellow line), for the years 1750–2008:

The main thing that the chart shows is that people have been using "advantage of doing" with some regularity in published (and presumably edited) writing for more than 200 years. Historically it seems to be roughly as common as "advantage in doing" and somewhat more common than either "advantage to doing" or "advantage from doing."

The earliest Google Books math for "advantage of doing" is from George Hutcheson, An Exposition of the Gospel of Jesus Christ According to John (1657):

Christ, having thus terrified them, doth now allure them to feed on him by shewing the advantage of doing it—to wit, that they shall have eternal life in the bud, and shall be raised up at the last day fully to enjoy it.

Given this record of long and steady use, I don't see why "advantage of doing" should be viewed as unidiomatic or somehow inferior to its rival forms.

On the other hand, placing the indefinite article an at the beginning of each phrase yields a somewhat different chart:

With the indefinite article included, "an advantage of doing" becomes less common than "an advantage in doing" and no better than equal in frequency to "an advantage to doing."

In this case, the earliest match is from Samuel Clarke, "Of the Necessity of Holiness," in Sermons, volume 7 (1731):

By the assistance of the grace of God, which he is always ready to afford to those that believe and desire to obey his Gospel, this temptation of great Prosperity may be overcome; and not only so, but even further, That Power and Riches which are to most men an occasion of falling, may become the matter of a more excellent Virtue, and an advantage of doing much more good in the world.

The expression continues to be used in serious texts, such as Goldman, Hein & Liben, Oxford Textbook of Palliative Care for Children (2006):

Acknowledging that a child is terminally ill is especially difficult in an intensive care environment. An advantage of doing so is that all diagnostic tests and therapies are directed towards a common goal: to increase the comfort and quality of life of the child (as opposed to the usual goal of striving to maintain life).

However, the phrase "there is an advantage of doing" is exceedingly rare, yielding zero Google Books matches. One published instance occurs in "Phone therapy better to treat depression," Chicago Tribune (June 5, 2012):

"Apparently, there is an advantage of doing therapy face-to-face, but the reason is not clear," Hofmann added in an email to Reuters Health.

I'm not sure why this wording hasn't caught on, but I agree with itsbruce's view (in a separate answer) that it doesn't sound very good. The seemingly similar wording "that's an advantage of doing" doesn't raise my hackles at all—and it draws three unique Google Books matches.