I think child's basis is the sum of:
For the gifted portion - it's either 50% of mother's basis if it would result in gain, or FMV @ time of gift if it would result in a loss, plus
cost of 50% of any improvements between the time of gift and DOD, plus
accounting for 50% of any other adjustments to basis between gift and DOD (one example: gov't eminent domain transactions), plus
the remaining 50% inherited portion @ FMV
I do also think that the mother's 50% would may need to go through probate. (edit: changed to "may", as Max said, depending on state law)