To show that it's "on Thursday".
See http://www.bbc.co.uk/wales/learnwelsh/pdf/welshgrammar_mutations.pdf , "Soft mutation" point 20.
(I think this may be from an earlier ar ddydd Iau "on Thursday", where the ar "on" dropped out in colloquial speech but the soft mutation caused by it remains.)
It's because dydd Iau is being used adverbially rather than it coming after a now missing preposition. That's why we say things like Dw i'n rhedeg bob dydd (I run every day) with a mutation on pob (every) as it's an adverb. The unmutated forms of ddoe (yesterday), wedi (after), gan (by) etc. aren't even found anymore as they're used adverbially so often.
Incidentally, we don't really use this soft mutation = "on" with days in the colloquial language. It's more normal to say Dw i'n mynd dydd Iau (I'm going (on) Thursday).
For much the same reason that it's Thursday and not thursday -- Iau (capitalised) is Jove or Jupiter, and stays capitalised also in Jupiter's day = Thursday (though English uses the Norse thunder god Thor rather than the Roman thunder god Jupiter in the name of its day).
Ah, sorry, I thought you were asking about the capitalisation of iau vs. Iau.
Your question is about the mutation vs. non-mutation of dydd Llun vs. ddydd Llun? (dydd Lun does not exist, though nos Lun does since nos is feminine and so lenites the following modifying noun)
ddydd Llun is "on Monday" -- the mutation shows that it is used as an adverb rather than a noun.
dydd Llun is "Monday" -- the day itself, a noun.
However, I believe that some areas in the south do not use mutation with dydd to indicate the adverbial use and so dydd Llun (without mutation) can also mean "on Monday" in those areas.
Yes, but ddydd Llun is mutated because originally "on Monday" was ar ddydd Llun - the mutation caused by ar (on). In speech the ar is often dropped because the mutation does the job of showing it is "on Monday".
So, ddydd Llun = ar ddydd Llun.