Alaikumassalam wa rahmatullah,
Ali bin Abi Talib has no daughter named Alizeh in the authentic Islamic sources, and just to make very sure I also looked at Shi`a sources, still no mention of this name or anything similar to it.
The only mention of a name sounding like this is in the Dehkhoda dictionary, the largest Farsi dictionary. The name is mentioned as a synonym for Alyas` (not Ilyas), a prophet mentioned in the Quran in 6:86 and 38:48, known as Elisha in the Bible. For this reason Alizeh can be used for a boy if by it you mean the prophet Alyas`.
Alizeh could also be a mispronunciation or variant of Alizaa, which is short for Alizaadeh, which means "offspring of Ali". Ali could be any person meaning Ali, though in Shi`a culture it refers to Ali bin Abi Talib. and zaadeh means "born", so it literally means "Ali born", born of Ali, i.e. a child of Ali. Since Ali means high, the name literally means "high born", "high status". I have never seen this used as a name however, it is only used as a last name in Farsi.
Alizé and Alizée, pronounced as Alizeh and Alizey respectively, is a French word meaning "trade winds", the winds that blow near the Tropics.
To sum things up, you can use Alizeh for a baby girl and by it you can mean "trade winds" (the French meaning). Or you can use it for a boy and by it you can mean prophet Alyas`. If Alizeh has other good meanings in other languages, you can use those meanings as well.