Your mum is right - I don't think people with hyperthyroidism can donate blood. Having said that though, if they are not currently taking antithyroid medications, and their thyroid levels are currently within the normal range (ie euthyroid), I think they should be able to, as not technically hyperthyroid,
Basically the thyroid gland is a gland in your neck which produces hormones which play a role in your metabolism. Hyperthyroidism means your thyroid gland is overactive - producing too many hormones and this leads to symptoms. There are a number of different reasons why this can happen.