Which java open source version supported by Mendix version 7.8.0?
AdoptOpenJDK should work. It’s also the new Mendix default JDK
Mendix 7.8 requires Java 8/JDK 1.8 (https://docs.mendix.com/refguide7/system-requirements), and for that you can use the open source AdoptOpenJDK implementation of OpenJDK 8 (https://adoptopenjdk.net/).
Andreas is right, AdoptOpenJDK is installed by default for the Desktop Modeler starting from Mendix 7.23.3.