I have a switch statement that looks at a variable value and based on that determines which data flow to execute. The problem with this is, I need to update the switch statement every time I add a new ID/dataflow.
Is there an alternative design to this? What if my dataflows had the same name as the variable value - would it be possible to parameterize the name of the dataflow to execute?
e.g. variable value = "1" execute data flow with name "1_dataflow", 2 execute "2_dataflow" etc. How would I accomplish this?

Yes, you can parameterize the values in any activity in Azure Data Factory and make the pipeline dynamic instead of giving hard-coded values.
You can also use
Set Variable activityin to define and variable and set value to it which you can use inSwitch activityand also can change it later.Refer: How to use parameters, expressions and functions in Azure Data Factory, Set Variable Activity in Azure Data Factory and Azure Synapse Analytics