My 2 cents. You don't have to make or give a gift, but a diamond sister spends so much time trying to make a new member feel comfortable, that even a thank-you letter means so much. Anything that comes from the heart is going to be well-received.
ADPi is moving toward (trying to!) get the chapter budget to pay for all gifts, so that the delta diamond sister is not burdened financially.
At my chapter, we had a policy of doing a gift exchange off-grounds so that diamond sisters could give each other gifts, yet no one would feel "left out" seeing if someone else got more gifts or bigger and better gifts than someone else. Our on-site gift traditions included:
1) At the Black Diamond Ceremony, we would present our diamond sisters with bookmark (and a bible provided by the chapter that was turned to a significant passage)
2) At Blue and White Day, we would group by Prides and this was a night where you would receive a framed poem from your delta diamond. The theme of the poem was "My Wish for You" and the delta would take a piece of existing writing or write her own thoughts. This also became a night for "pass down" gifts.
3) At initation, after the ceremonies, we would have cake and then we would go outside where there was a huge diamond on the ground formed from many diamond shaped pieces of wood. Each delta diamond sister was provided with this diamond board by the chapter and she decorated it for her alpha diamond sister (fabric, wood letters, paint, etc.). Typically, off-site, the delta diamond sister would also give her alpha diamond a letter-jersey shirt, a lavaliere and other gifts like alphie lions, ADPi novelty gifts and homemade items with letters on it. The Alpha Diamond would usually give a wood gift to the delta, and maybe some other homemade ADPi gifts, too.
When I went alum, I became very close with an alpha who pledged the following fall. While she was in initation, I had her roommates let me into their apartment and I decorated her room with posters and blue and white streamers. I had a big box of gifts and a card waiting for her when she got home. Oh and this is the best part! I created a bunch of little poems that were written on cards throughout the apartment, and you were led in a straight path with little pawprints I had made-- she loved it!

To this day, we refer to each other as Adopted Diamonds (in our chapter, an alpha and delta who weren't diamonds might become very close and they would unofficially call each other Adopted Diamonds-- or you would call them "Cubic Zirconias" or CZ's!)