In Hebrew "aliyah" is not a verb but a noun that is often translated as "ascent" (or "the act of going up") and has often been used to refer to Jewish migration to Israel.
But the use of Aaliyah as a female name actually comes from the Arabic. As an adjective (or, technically, an active participle), the word may be translated to English as "high", "exalted", "honorable", "remarkable", etc.
This is probably the best explanation I've heard of the Arabic-language origin and meaning of both the word and the name:
In light of Ms. Edwards' commitment, I really like what he says toward the end:
"If you carry the name Aaliyah, you have a responsibility, so to speak, to live up to your name by doing something outstanding, something remarkable. Your name in itself is a calling to be greater than the norm, to work out whatever is the average around you and to be better than that average. So in every aspect of your life, some scholars of language write, the person who carries this name should aim to be aaliyah in every single facet of human life. ... The one who carries this name should aim at all times to be exceptional, to be remarkable."
May it be so.