What is the meaning of "I would argue that" and "arguably"?

Both I would argue that and arguably are examples of hedging. In The Sense of Style Pinker states in his section on hedging (p43):

Many writers cushion their prose with wads of fluff that imply that they are not willing to stand behind what they are saying, including ... the ubiquitous I would argue (does this mean that you would argue for your position if things were different, but are not willing to argue for it now?).

Of course, the answer to Pinker's rhetorial question is No. I would argue is simply a way to state your position more tentatively than coming right out and saying, as in the present example, It was the greatest rain of the century.

Arguably is also common hedging device, used by writers who wish to avoid making definitive statements. But arguably can mean different things in different contexts. This is well explained by Peters in The Cambridge Guide to English Usage (p46):

There's a latent ambiguity in arguably as to whether one is arguing for or against a proposition. The affirmative use is often spelled out by an accompanying superlative or evaluative expression, as in arguably the most powerful package, arguably a hazardous occupation, arguably the buy of the season ... . The word allows writers to have it both ways, to say that a "case can be made out" without actually committing themselves to it. The equivocation takes over in some instances, as in what is merely arguably right, and the word comes closer to its negative use "capable of being disputed". But whether distanced from or closer to a given point of view, arguably leaves the advocacy to someone else.

Since the present example, It was arguably the greatest rain of the century, contains a superlative, the sentence can be interpreted affirmatively as something like:

I believe that it may have been the greatest rain of the century, but others may disagree, so don't quote me on it.

Merriam Webster's Dictionary of English Usage (p116) has a long article about arguably, tracing its rise in popularity and its particularly common usage in sports reporting. For example:

• Arguably the greatest European sprinter of all time
• Arguably the greatest skate racer ever
• Arguably the country's leading authority on college football

Merriam Webster notes that some self-appointed arbiters of good style have called the word modish, grossly overused, and unnecessary. But it comes to the sensible conclusion that the word fills a need in the language.

There is a good article about hedging from the University of London: Hedging in Academic Writing.

And there are other questions about 'arguably' elsewhere on this site.