I always offer little foraging bins and they seem to really love them.
You get a little tupperware tub (the type used for brownies or cupcakes, so shallow sides) and fill with timothy hay and little treats. I mix it up in terms of the treats. I've used tonic seeds, crushed freeze dried bugs, Miracle Meal. Anything that won't "go off" if they leave it in there for a day or longer.
You can also put little stones on the bottom of the bin. I'd done this originally to add weight to help keep the bin in the center of the cage to limit the mess. But I've found that the little stones added an element of complexity to their foraging, which they enjoyed. (e.g. the treats would get caught under the little stones, so they'd have to nose around a bit more to get them.)
They love rooting around for the treats and they love playing with the bits of hay, carrying them around and starting little nests.
I think that fulfills some of their nesting urges too.
I've tried toys with bells and mirrors, but the finches have never really used them.
I do think that enrichment is a huge part of finch keeping. Many people think of them as simplistic or "easy" birds, since they're not as demanding of humans. But they do have complex needs socially and mentally. Enrichment is a very important part of the equation. Otherwise, they get bored and when they're bored, aggression or other destructive behavior emerges.
I tend to agree with
Sheather that it does seem wrong to deprive certain birds of a nest. This is especially true once you see how much they enjoy building and un-building and re-building it. It clearly gives them purpose and joy. And it's easy enough to pull out eggs if they're laid.
I've found that birds who get too aggressive over nests can do well with the components of a nest. (e.g. a bird shelf in the corner and some hay and bits of toilet paper). They'll set to work trying to build a nest on the shelf. They don't seem to be as possessive of these nests-in-progress and most are woeful nest builders and don't get much further than a loose pile of hay and paper. (Though they do learn and improve over time.)
But it doesn't seem to matter to them. It seems that most of the joy is in building the nest, not really using it. (Evidenced by the fact that I have several who are perpetual nest builders. They're always stuffing nests, then un-stuffing them, then re-stuffing. So in our bird room, it's either a building day or a tear-down day! LOL)
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