Flowers could be nice, since carnations do last longer than other flowers. Jewelry-wise, bigs in my chapter typically only give lavaliers, if anything, so depending on the norms for her chapter, jewelry could be a safe bet. Maybe you could get in touch with her big so you don't buy something that her big already bought her. I have no idea what you have in mind, price wise, but from piphiexpress, if I were a Pi Phi, I would want these:
Arrow Ring, $100 (under official jewelry)
Silvertone Wing, Arrow, and Heart Necklace, $15 (under fashion jewelry)
Arrow Tote, $12
I just had to post a picture of this badge. Everyone ooh and ahh.
ETA: Don't sweat that you can't come to initiation. Initiation is ritual, so you wouldn't be able to see her initiation anyway, barring some norm at this campus that parents come afterward(??). The only time I've ever heard of family coming is if a legacy is being initiated, in which case, the legacy's member relative would attend.