The origins of the words "Buda" and "Pest" are obscure. According to chronicles from the Middle Ages the name "Buda" comes from the name of its founder, Bleda (Buda), the brother of the Hunnic ruler Attila. The theory that "Buda" was named after a person is also supported by modern scholars. An alternative explanation suggests that "Buda" derives from the Slavic word "вода, voda" ("water"), a translation of the Latin name "Aquincum", which was the main Roman settlement in the region.

There are also several theories about the origin of the name "Pest". One of the theories claims that the word "Pest" comes from the Roman times, since there was a fortress "Contra-Aquincum" in this region which was referred to as "Pession"  by Ptolemaios.

According to another theory, "Pest" originates from the Slavic word for cave "пещера, peshchera" or from the word for oven "печь, pech'", in reference to a cave where fires burned or to a local limekiln. In the old-Hungarian language there was a similar word meaning oven/cave and the original old-German name of this region was also "Ofen". Later, the German "Ofen" referred to the Buda side.

Learn more about Budapest and Hungary from the source: wikipedia.org