This has vexed some for quite awhile now. As one columnist opined in 2009:
When the 10-year span starting with 2000 began, there was much consternation over what we would label the period, a convention that has become necessary for things like nostalgic miniseries titles and radio flashback weekends.
The Zips? The Zilches? The Naughts? The Aughts?
The Double-Os? The 2000s?
All of them had flaws and none, as the decades closes, has stuck.
(Steve Johnson, Chicago Tribune, April 14, 2009)
That said, back to your question:
But what if I make a pun over 20 years, that is funny now, but won't be in future? What do I say? "You won't get it, it's __ thing."
It's worth pointing out that, if you don't like Naughts, Aughts, Zips, or Zilches, you're not required to refer to a decade numerically. How about:
You wouldn't get it; it's a millennial thing.
At least one dictionary lists, as a definition of millennial:
A person from the generation which grew up in the 90's and 2000's.
Many people in education use this term to refer to learners who grew up during the age of the internet. They don't remember a time without cell phones, email, and social networks, because those things had proliferated by the time they were early enough to start forming memories.
The millennial generation has come to college. This is a new breed of students likely to challenge paradigms in higher education. Howe and Strauss (2000) affirm that millennial students started entering college in 2000. Thus, it becomes relevant, given their particular characteristics, to address the challenges that this new generation pose to higher education.
(Rivera & Huertas, Millennials: Challenges and Implications to Higher Education, 2006)
So, twenty years from now:
You: Oh, man, that is so funny! It's like, “Dude, your mama's on Facebook now!”
Your young nephew: Huh? What are you talking about?
You: Sorry, you wouldn't get it. It's a millennial thing.