Records? What type of records?
Am I correct to assume your Syrian family were Christians? Most Syrian and Lebanese descendants I know in Cleveland are Christian.
The church would be the best place to begin, as often times the churches were the official keepers of birth, marriage and death records.
Here is a link to a tutorial regarding research in Syria (which given the present conditions may be difficult)http://www.genealogytoday.com/family/syrian/part5.html
Links which may be of interesthttp://www.rootsweb.ancestry.com/~syrwgw/http://www.aina.org/ata/20101028214412.htm
Also I notice the Lebanon board here at Genforum has many queries re Syrian genealogy. And Ancestry.com has a board dedicated to Syriahttp://boards.ancestry.com/localities.mideast.syria.general/...
Croatian Heritage Museum