mother inherits house and gets step up to FMV. she gifts 1/2 of house to her son who gets 1/2 of her basis. when she dies and the son inherits her half. his basis would be 1/2 the original basis he had from the gift + 1/2 the FMV on the date of his mother's death.
so his basis would be 1/2 the $450,000 and 1/2 the $700,000 if that's the fair market value on the date his mother dies. if he sells for the $700K his gain would be $125K less s selling expenses.
see a lawyer but it seems the son would have to be removed from the title which would likely constitute a gift to his mother for which a gift tax return would be required. There would likely be costs involved which the lawyer could explain to you. for example, if the son is removed from the title, ownership of the house would likely need to be probated.
