Claiming mind is the mind that claims things to be "mine." "My" this and "my" that. We tend to claim a lot of things to be "mine." The senses come in contact with an object, there is a feeling and then we jump on it, we claim it and we want it to be a certain way, that is the claiming mind. We've made it into "mine." But when we examine life as it is, we see that nothing is really "mine." We are attached to experience and formed it into "mine."
As far as Buddhist practice goes, it is very important to see how we make everything into me and mine because this is the arising of the ignorance. As we start to see the suffering that comes from the claiming mind and have enough compassion for ourselves, we tend to let go of the idea that everything is mine and has to be a certain way. We are more able to see life clearly as it is.
This is the key to letting go of suffering. There is the contact between the senses and their object and the arising of feeling - pleasant, unpleasant, or neutral. Then there is the reaction - like, dislike, or indifference. Usually then there is the claiming, the making of things into mine. This is called "birth," and because of birth there is death, aging, sorrow, lamentation, pain, grief and despair, This is the arising of all suffering. In essence, that is a short description of the teaching called, Dependent Origination. This is what the Buddha was contemplating deeply on the night of his enlightenment. So by becoming more aware of the senses and the contact, and more aware of the Vedana and the urge in the mind to be "born," to make things into me and mine, we have the opportunity to let go of the cause of suffering.
As we understand impermanence more deeply, we understand that things are arising and passing due to cause and effect and that it's quite an impersonal process; that nothing really is mine. So the claiming mind, yes, we have to be aware of this claiming mind, if we are going to end our Dukkha.