I think the "something cheap" here is Bowie (a rich man) being humbled by the death of his special friend.
Being humbled he feels that making an expensive grandiose gesture would perhaps make a false impression when what he wants is to be as honest and direct as he can in his mourning.
Perhaps the picture will serve as a shrine, and Bowie wants that when he speaks to his friend his friend understands that his grieving is sincere and not for show.
When poeple see the frame they will understand immediately that it is not a piece of furniture to increase the beauty of his home.
Also the simplicity of a cheap frame emphasizes the importance of the picture for a wealthy man.
Perhaps he knew the friend when he was poor, and in remembering those times he feels that only simplicity itself would do justice to the purity of their relationship, ie the friend was not a typical sponger interested in him only for his money, but knew and liked him when he was nothing.
Perhaps the friend was a true friend, a soulmate, as opposed to being someone he knew by virtue of his status or class.
Bowie is being anti-materialistic for the wealth of the friendship he has lost is far greater than material possessions.
Contrary to how it seems this is a true friend he is mourning.