I would think the problem is not so much that the donor is a non-member but that the donation is not for an allowed purpose. Ward activities are paid out of budget funds and donations to budget aren't allowed. But I expect they don't want to offend the donor by refusing the donation.
Depending on the situation there might be some possible options. Perhaps the donor would be happy to have the funds used for YM and YW camp (although even there the rules for funding camps don't mention 3rd party donations other than in connection with a fund raiser). Or perhaps the ward could have a service activity and use the funds for things related to providing the service (so the donation doesn't fund the activity but rather the service). In any case, it would be up to the bishop, in consultation with the stake president, to determine whether this is an appropriate and allowed use for the funds.
On the more technical side, if the bishop decides the donation can be accepted and used, the money should go into an appropriately name Other account for whatever specific purpose is decided on. It should not be deposited in a Budget account.