BEITIA the meaning is DOWN, but in toponims and surnames is "The house of the Down part", and in many surnames is used as a sufix. I think the origin of the surname is Bizkaia, there are a lot of baserriak with this name (Kortezubi, Fruniz, Gorozika,Ispazter, Ibarrangelu, Abando,Arrazola, Bedarona, Bermeo, Manaria, Mendata, Morga, Mungia, Zenarruza, Erandio, Rigoitia......) There are a few in Gipuzkoa (Albistur, Itsansondo, Mendaro), so the only way to know the origin of your surname is to make a good genealogical Tree. The opposite to Beitia is Goitia.

There were Beitias Knights of Santiago in 1688,1692,1784, and 1803, and Knight of Carlos III in 1820

Coat of arms: I have found 4, depending of the place