How can our department solicit more (good) PhD applications?
You could kill two birds with one stone and target underrepresented groups.  Make connections with undergraduate student organizations of color.  Invite undergrads taking junior level classes in your department to meet with your committee at a barbecue to talk about what they have found most helpful in terms of supporting their academic success.  If you ask the question that way, you may get some helpful constructive feedback, without actually inviting them to kvetch.

Make sure your department has an effective community outreach program to K-12 students, and ask them to target underrepresented groups in your town or city.  Word will spread.

Co-sponsor, with the engineering department, and a student organization, a showing of the film "Underwater Dreams".  Better yet, invite the people involved in making the film to your showing, for a panel discussion after the movie.

Make sure your faculty members' webpages are clear and inviting.

Recruit some peer advisors and give them some online profiles.  Make sure the peer advisors have a quick way to get answers from the members of your committee in case there's something they can't answer.

I don't know if women are an underrepresented group in your field, but even if they aren't, make sure you have all the features that make a department women-friendly.

With respect to the GRE comment -- I don't have an opinion about this suggestion, but at least try to come up with an admissions policy that looks at the whole person, and make that policy clear to potential applicants.  A public statement along the lines of "we encourage returning students to apply" is a helpful clue to women who have taken a child rearing break.

Make sure you have an undergrad and grad student prize in your department, for example best poster, best thesis, and really celebrate the winners.  Feature different student bios on your department website on a rotating basis.

Get some funding for some upper level undergrads to go along to a conference.  Set up peer mentoring to make connections between students of different levels.

Make sure there are fun things for students to do together in free time, e.g. soccer or volleyball or Eurogames or whatever your department's students enjoy doing.

Really nice administrative support personnel can do a lot to make a department attractive to students.  (For example, "if you have any trouble getting registered for a class you need, let me know".)

Sorry if that was a bit disorganized -- hope these ideas get you thinking.