Since the word is "would," not "will," instead of trying to wade through that OED link's lengthy treatise on "will" from the 1989 edition, try this link to the current OED definition of "would" through Lexico:
- ironic Used to make a comment about behavior that is typical.
‘they would say that, wouldn't they?’
The OED's example above can just as well be simplified to:
They would say that!
Notice the similarity to your sentence:
It would have to rain today, of all days!
That could just as well be expanded to:
It would have to rain today, of all days, wouldn't it?
So, the definition of "would" in your sentence is essentially "is so typical for it to" (i.e., "It's so typical for it to have to rain today, of all days!"). "Typical" is being used in accordance with its following definition:
1.2. informal Showing the characteristics expected of or popularly associated with a particular person, situation, or thing.