Budjawi is what locals call the fruit of the buri or century palm tree. Boiled budjawi are sold (around PhP10 per cup) at the Central Market but I had been unlucky everytime I went there since there wasn't even a single vendor selling them. Luckily I found these photos from another good food blog BUCAIO. 
To eat budjawi one has to "break apart" the fibrous outer shell. And I quote Kai "The outer cover is fibrous, but the fruit inside is white and chewy - very, very fun to pop out and chew. It has the texture of a slightly undercooked sago (pith palm pearls, or "bubbles") so is slightly unyielding, with the subtle flavor of water chestnuts, or even toge".
Acknowledgments to Kai's bucaio blog for the photos and text snippets.