This is a word we borrowed from the Spanish word for "table," mesa. In Spanish, "the table" is "la mesa." Our mañaina pronounced it "la masa" and that's where we got lamasa. So, yes, the Mesa family has "table" for a family name.
To "set the table" is plånta i lamasa.
Pale' Roman de Vera, Capuchin, always trying to get back to an indigenous term, though he was a Spaniard himself, offers us the word fañahangan for "table."
Remember the FAN+WORD+AN formula? It makes a word become the "place of" or the "time of" that word. Sahang means "to place something above the floor." A table does exactly that.
FAN+SAHANG+AN = fañahangan.
It would be nice to find out if Pale' Roman got fañasahang from an older Chamorro or missionary who remembered it as another word for "table," or if he read it in an older book (less likely) or if he just used logic to come up with the word.