In my humble opinion, I think that a lavalier would be the wrong way to go. Instead, I think a "Friendship Pin" would be more appropriate. As afar as I know, a lavalier isn't an official recognition symbol of the fraternity. A far better choice for you to present your female friend with, would be the piece fo jewelry found
here.
(or cheaper
here.)
The "goal" that a lot of women seek in our jewelry (if I can be so modest for our bond), is to obtain an official brotherhood badge. Short of proposing to this girl, she can't get one of these from you. Instead, here's an explanation of the badge that would be much more appropriate for you to give her:
The Friendship Badge
The cross and crescent with shield bearing the Greek letters pin was originally designed to be worn by girlfriends of members when the relationship had not yet reached the state of engagement or marriage. Today, the pin also is worn by members who have not purchased a badge, or by members who prefer its design.
and (another explanation from another website)
The Friendship Badge, created by Jack Mason following controversy over whether sisters, housemothers, and steady girlfriends could wear the badge, is a crucicrescent with a shield superimposed and the letters of the Fraternity written onto it. Created by Balfour, it “carries no promise of protection from the Fraternity, and is simply a casual compliment. It does not require the salute of removing one’s hat…”
So, it's a bit more casual than the badge you give to a wife, but it still symbolizes your affection to your female friend. I'm sure she would appreciate this badge a great deal. Since I have not proposed to my girlfriend (yet?), I have not given her a badge. Instead, I gave her the friendship pin, and she wears it proudly. The lavalier can come later...like I said, it's not an official LCA piece, so it doesn't carry as much value as the previously discussed ones.
Good luck!
Yours in Z A X ,