Q:

What was Adolf von Harnack's case that Priscilla wrote Hebrews?

One of the theories about the authorship of Hebrews is that Priscilla (the wife of Aquilla) wrote Hebrews.  While it is by no means universal (indeed many suggestions have been made), and at least in my estimation not even the most persuasive case, some have suggested female authorship for this epistle.
What factors, other than simple desire on the part of some that there is a female author in the Bible, led people like von Harnack to believe that Priscilla was the author.  While the wikipedia article mentions why a female name would have been blotted out, it doesn't really point to any evidence for the case.  What textual clues led von Harnack to even make the case? 

A:

The authorship of Hebrews cannot be known for certain, but we can know several things about the author.

The early church counted Hebrews among the letters of Paul, even though it does not bear Paul's name. Clement of Alexandria and Origen noted that the writing style did not match Paul's, but did not consider it a forgery. This indicates that they believed Hebrews contained teaching from Paul and likely was written by someone who traveled with him.
The author quotes extensively from the Septuagint (the Greek translation of the Jewish Bible) and has an extensive Greek vocabulary. This indicates a highly educated Jew who was also a Roman citizen.

The Book of Acts provides us with several names that fit this profile:

Barnabas, a Levite from Cyprus (Acts 4:36)
Apollos, an Alexandrian Jew (Acts 18:24)
Aquila, native of Pontus (Acts 18:2)
Priscilla
Luke, the author of Acts

The original intended audience of Hebrews also may provide clues as to its authorship. Harnack (among others) suggested the original recipients were in Rome, for the following reasons:

Of all Paul's letters, Romans is the most similar to Hebrews in terms of its treatment of Jewish law and customs.
Second century Roman Christians quoted Hebrews much earlier than Christians in other locations.
The mention of the recipients' previous suffering, persecution, imprisonment, and confiscation of property (Hebrews 10:32-34) appears to be a reference to Nero's persecution of Christians after the fire in Rome.
The reference to "those from Italy send their greetings" in Hebrews 13:24 Harnack took to refer to Christians who had been exiled from Rome by Claudius (Acts 18:2)

If the letter was addressed to Roman Christians, Priscilla and Aquila are the most natural candidates for authorship. They had hosted a house church in Rome (Romans 16:3-5).
They were also known to have "explained the way of God more fully" to Apollos (Acts 18:24-26). Furthermore, the author of Hebrews sometimes uses the pronoun "I" and sometimes "we", which would make sense if Priscilla and Aquila both had a hand in organizing the material.
Finally, Harnack argued that because of the lower status given to most women in ancient times, Priscilla would have had a greater incentive to have written anonymously than any of the other possible authors. (Or alternatively, that the original recipients would have had a greater incentive to delete her name from the beginning of the letter before distributing it.)

A:

I have researched and written about the case for Priscilla's authorship of Hebrews. Your summary of Harnack's arguments is good. Please refer to several of my published articles on the following website: www.clarksons.org/spiritleads/ruth_hoppin.htm. My book, "Priscilla's Letter" ("La Carta de Priscila") has even more details. As for the "masculine participle" in Heb. 11:32, the neuter, identical in form, was likely intended, as customary when persons in general are referred to.
Ruth Hoppin

