What is the source for making barbecues on Yom Haatzmaut?
SimchasTorah pointed to a fascinating comment from Meshech Chochmah (Rabbi Meir Simcha of Dvinsk, c. 100 years ago), let translate the entire quote:
... therefore, the fact that the Jews would [celebrate Passover by] eating flame-roasted lamb would not raise any questions, not even for the wicked son; as it's perfectly appropriate to celebrate a national holiday and make a day of feasting and joy, a day on which they left the slavery of Egypt, gained their freedom, and became a great nation of distinction. But why this requirement to first apply blood to the altar [and the doorposts in Egypt counted as an altar because G-d's presence was felt there] before eating the meat? That makes people ask, "what's this worship of yours?" And you answer -- "this is a Passover sacrifice to G-d"; His presence and glory was revealed to us in Egypt -- that is a lofty spiritual matter that was unheard of in other nations.
The Meshech CHochmah on Bo in the Pargragh starting Ushmartem 11 lines in it says that the proper way to celebrate is to eat roasted meat see it for yourself
Adraba! There is no source. It is a natural expression of the inner yearning of Am Yisrael to reestablish our broken relationship with the Ha'Kadosh Baruch Hu with the Avoda of Korbanot in Beit Hamikdash.
Thanking Seth J for his comment below I'd like to clarify my original answer a bit.
It should be clear that BBQs on Yom Ha'atzmaut are not and should not be performed Zecher L'Korban. If done is such a manner it may even create a problem with eating the meat in terms of the prohibition of Bamot.
With that said, the setting of a BBQ is very similar to the setting when bringing and feasting on a Korban Toda. Therefore we could look at this modern day minhag as a (spiritually) natural development. Getting us prepared. The original question was about a source. My answer was that I don't see a need for a source. Even if a source is found it is most likely not what started the minhag. The minhag is already a few decades strong. And ... if you like juicy sizzling meat, it is a great minhag!