Alaikumassalam wa rahmatullah,
There is no Arabic/Quranic name that sounds like Elizah. The name can be formed from the Arabic word Ali (high) and zah (born), which would create the word "high born", i.e. noble, high status, a person who comes from respected family. I haven't seen this used as a name, but it is grammatically correct and has a good meaning, for this reason you can use it for a girl, whether you spell it as Elizah or Alizah.
The name of Maryam's aunt as mentioned in al-Qurtubi (Islamic historian) is Eeshaa`, not Elizah. Christian sources say it is Elizabeth, which can be shortened to Eliza. Eeshaa and Eliza might be the same person, but since the name Eliza is only mentioned in Christian sources, some scholars wouldn't recommend using it (the same way Prophet Yusuf's brother's name, Benjamin, is not used as a name by Muslims since the name itself is not from Islamic sources).
For this reason it is better if you use Elizah with the "high born" meaning above, instead of using it as short for Elizabeth, which is a Christian name.
You may also find this answer interesting: Meaning of the name “Alizeh”