What were the names of Samuel's sons, whom he appointed as judges in Beersheba?

Answer: Joel and Abijah

Samuel's firstborn was Joel and his second Abijah (KJV: Abiah); they judged in Beersheba but did not walk in their father's ways.

“When Samuel grew old, he appointed his sons as judges over Israel. The name of his firstborn son was Joel, and the name of his second was Abijah. They were judges in Beersheba. But his sons did not walk in his ways; they turned aside toward dishonest gain, accepting bribes and perverting justice.”— 1 Samuel 8:1-3 (BSB)