Isaiah was active as a prophet of Judah living in Jerusalem from 740–701 BC. Lehi and Nephi left Jerusalem approximately 100 years after Isaiah’s death. This may sound like a long time later, but if you consider it in the context of generations, it is only two to three generations later. When Lehi was a young boy, there may have been people alive who remembered the Prophet Isaiah, just as there were still people alive in late pioneer Utah, in the early 1900s, who could tell children what it was like to listen to the Prophet Joseph Smith. Nephi likely thought of Isaiah as The Prophet, much as we think of Joseph Smith as The Prophet. Isaiah was that important.
David Rolph Seely, "Nephi’s Use of Isaiah 2–14 in 2 Nephi 12–30" in Isaiah in the Book of Mormon, ed. Donald W. Parry and John W. Welch (Provo, UT: FARMS, 1998), 151.