My mother will be washing my clothes.
My clothes will be being washed by my mother.
Is this correct? It feels very odd to use 'be being' together like this.
'My clothes will be washed by my mother'. I'm not aware of any other possible construction.
Even had you added an extra clause to the first sentence ('My mother will be washing my clothes next Sunday when I come to visit'), the passive will still be 'My clothes will be washed by my mother next Sunday when I come to visit'.
The problem is most likely with the Active voice version, not the conversion to passive voice. "will be washing" is future continuous: you normally only use this if you want to describe two things that are happening at the same time, for example:
I will be lying in the sun while you are stuck in the office
So, unless you particularly want to specify something else that is happening while your mother washes your clothes, it would be better to write the first sentence as
My mother will wash my clothes
And then make the passive voice version like this:
My clothes will be washed by my mother.
If you really did want to specify something that was happening at the same time:
My mother will be washing my clothes while I am out partying
Then the passive voice version would be
My clothes will be being washed by my mother while I am out partying