Is Isaiah mentioned by name in the New Testament?

Yes, many times.

Matthew 3:3, 4:14, 8:17, 12:17, 13:14, 15:7

Mark 1:2, 7:6

Luke 3:4, 4:17

John 1:23, 12:38-41

Acts 8:28-30, 28:25+26

Romans 9: 27-29, 10:16-20, 15:12