While uncountable nouns don’t have plurals (usually, although there are exceptions to that rule too) they can in certain circumstances be preceded by an indefinite article. These circumstances are when you are qualifying or limiting the noun’s meaning in some way.
In these cases, the uncountable noun’s meaning is definitely limited by an adjective or a subordinate clause:
Another example can be found here:
The indefinite article is used with uncountable nouns in order to suggest the idea of ‘a kind of’.
This might also be true speaking of uncountable nouns used without any determination by an adjective or a subordinate clause.
The one with the indefinite article seems to be limiting the meaning of the noun. In other words, we have an omitted (presumed) attributive there: What a [fantastic] patience! What a [terrible] weather!, and that seems to be adding a kind of emotional emphasis. However, I’m not quite sure about that, and I fail to find any reliable sources. So, any hints would be really appreciated.