You're looking to do a Pollyanna.
The name itself is neutral though it is ultimately an obscure allusion to a 1913 story about a girl named Pollyanna who got crutches for Christmas when she wanted a doll. (And the very same girl who lends her name to "Pollyanna" meaning "an excessively cheerful or optimistic person" as per NOAD.)
Pollyanna is a somewhat general term, and is used for both Secret Santas and white elephant style exchanges and also exchanges where who gets what gift is decided after all the gifts are bought and wrapped. However, in every single one I've seen, the organizers set up some ground rules, especially about what types of gifts can be given.
Here are some examples in use:
As you've read, the spirit of this game is about finding the good things in life, no matter the situation, finding a gift to exchange with others that has value, that speaks to a person's positive personality traits. Each guest's goal is to find a gift that is thoughtful, that takes into account who the receiver is or might be and what you love about them, or to reinforce or remind the receiver about something good that this person has done in their lives.
There are a few variations of the rules to playing this game, from buying a gift for a certain person within the group to having a random guest choose each gift. Both variations work well and are described later on in this article under Pollyanna Gift Exchange Rules.
Pollyanna Gift Exchange (archive 1, archive 2)
After Christmas, but still over the holiday break, we do a “Pollyanna.” This is when everyone in my extended family picks a name, and then buys a gift for that person. A few days after Christmas, we meet and exchange gifts.
PhillyBurbs: Christmas traditions (by a high school student in SE PA)
Although everyone likes to get a bunch of presents for Christmas, like many families, ours is expanding to the point that exchanging gifts between adults is just cost prohibitive and somewhat silly. As such, we all buy gifts for the kids, but the adults do a Pollyanna in which everyone enters their name into a bowl, and each adult draws the name of one person for which they’ll buy a small gift for.
Inexpensive pocket transit (by a user in NE PA)
If you've never heard of this term before, that's probably because it's regional. The heart of the region where it's used seems to be Pennsylvania, where some people are shocked to find out it's not used elsewhere.
See also the research done by World Wide Words into this expression.