If you are running it as an embedded database in spring I use the following configuration to enable the built in web client when the main app is running:
<!-- Run H2 web server within application that will access the same in-memory database -->
<bean id="h2Server" class="org.h2.tools.Server" factory-method="createTcpServer" init-method="start" destroy-method="stop" depends-on="h2WebServer">
<constructor-arg value="-tcp,-tcpAllowOthers,-tcpPort,9092"/>
</bean>
<bean id="h2WebServer" class="org.h2.tools.Server" factory-method="createWebServer" init-method="start" destroy-method="stop">
<constructor-arg value="-web,-webAllowOthers,-webPort,8082"/>
</bean>