Recommended prerequisite: check which versions of Java may already be installed using the following command: /usr/libexec/java_home -V
The following commands worked for me:
brew search openjdk # check if desired adoptopenjdk8 cask may already available
brew search java # (optional)
brew tap AdoptOpenJDK/openjdk # (assuming that the desired adoptopenjdk8 cask is not already available)
brew search openjdk # check that the desired adoptopenjdk8 cask is now available
brew install --cask adoptopenjdk8
Then the following commands confirmed that I was using the desired OpenJDK 8 version by default (with no JAVA_HOME environment variable setting):
java -version
javac -version
And the following commands confirmed that I have both Java SE 8 and OpenJDK 8 installed:
/usr/libexec/java_home -V
ls /Library/Java/JavaVirtualMachines
Note that which java and which javac show that java and javac are still run from /usr/bin.
Followup step: remove the old Java SE SDK from /Library/Java/JavaVirtualMachines, as documented in: https://docs.oracle.com/javase/8/docs/technotes/guides/install/mac_jdk.html#A1096903
AdoptOpenJDK 8 package notes May 2019: adoptopenjdk8 was also added to homebrew/cask-versions, and this led to some issues: https://github.com/AdoptOpenJDK/homebrew-openjdk/issues/106
There is now a proposal to rename adoptopenjdk8 to openjdk8 in homebrew/cask-versions: https://github.com/Homebrew/homebrew-cask-versions/pull/7375
If this proposal is merged, this answer will likely be updated to recommend installing the renamed cask.
See also (some references):