Yes you can definitely do in Weblogic, Websphere, Jetty & Tomcat prior to 7 (because I have done it). But the Java Servlet API up to Version 2.5 states that the session identification cookie must be named JSESSIONID
weblogic.xml
<session-descriptor>
<cookie-name>myCustomSessionId</cookie-name>
</session-descriptor>
Jetty
The Session Management of Eclipse Jetty allows for setting both the session cookie name and path parameter name via either WEB-INF/web.xml
context parameters, or via init parameters on specific contexts, or even on the server side Session Manager (to apply this setting to all deployed webapps on the server).
Outlined in the Session Management documentation.
<?xml version="1.0" encoding="ISO-8859-1"?>
<web-app
xmlns="http://java.sun.com/xml/ns/javaee"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd"
version="2.5">
...
<context-param>
<param-name>org.eclipse.jetty.servlet.SessionCookie</param-name>
<param-value>XSESSIONID</param-value>
</context-param>
<context-param>
<param-name>org.eclipse.jetty.servlet.SessionIdPathParameterName</param-name>
<param-value>xsessionid</param-value>
</context-param>
...
</web-app>
Jetty also supports the Servlet 3.0 session-config name configuration
<?xml version="1.0" encoding="ISO-8859-1"?>
<web-app
xmlns="http://java.sun.com/xml/ns/javaee"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_3_0.xsd"
metadata-complete="true"
version="3.0">
<session-config>
<comment>This is my special cookie configuration</comment>
<domain>foo.com</domain>
<http-only>false</http-only>
<max-age>30000</max-age>
<path>/my/special/path</path>
<secure>true</secure>
<name>FOO_SESSION</name>
</session-config>
</web-app>
Tomcat - context.xml
<Context path="/myApp" sessionCookieName="myCustomSessionId">
Tomcat no longer accepts non-specification compliant name-only cookies by default. However, a new system property has been added, org.apache.tomcat.util.http.ServerCookie.ALLOW_NAME_ONLY, that can be used to accept name-only cookies.
IBM Websphere 6.1
Servers > Application servers > server_name > Web container settings > Session management > Enable cookies
Cookie Name - your new name