Helper function to return a string based on passed object and some defined rules

Some simple pointers...

• You have boolean assignments that are only used once, for conciseness it may be preferable to inline them:

  // old
  boolean isUpdateMessage = (chat.getTag().equals(TAG_UPDATE_MSG));
  if (isUpdateMessage) 
  
  // new
  if (chat.getTag().equals(TAG_UPDATE_MSG))
  

• Not sure if the getter methods on your ChatMessage class can return nulls or not, but in the unlikely event they do, you may want to practice some defensive coding by switching the .equals() around, e.g. if (TAG_UPDATE_MSG.equals(chat.getTag()))
• Your comments in the code code snippet are redundant, can consider removing them too for conciseness too.

edit: (prompted by OP's first comment)

Wells, this shouldn't be recommended on CR but just for fun, I tried golfing your method and reduced it to a few ternary operators as a 'one-liner':

return chat.getTag().equals(TAG_UPDATE_MSG) ? 
    Resources.string(UserRelationshipInfoManger.getInstance().getInfo(chat.getUserId()).getSource() == SOURCE_FRIEND ? 
        R.string.text_user_is_friend : 
        R.string.text_user_is_stranger) : 
    chat.getTag().equals(TAG_TEXT_MSG) ? 
        new TextInfo(chat).getText() : 
        String.format(Resources.string(chat.getType() == DIRECT_MESSAGE ? 
            R.string.text_new_direct_message : 
            R.string.text_new_message
        ), chat.getUserInfo().getDisplayName(), Resources.string(getMessageType(chat.getTag())));

The only assumption I made is that the relationshipInfo will not return null, so I can eliminate the null-check when assigning isUserMyFriend. Once I got to this golfed version three minor improvements become clearer:

1. You can consider a supplementary method isUserMyFriend() in the UserRelationshipInfoManger instance to give you a boolean value for checking if the user is a friend or not.
2. You can consider a supplementary method getUserDisplayName() in the ChatMessage class that simply calls getUserInfo().getDisplayName().
3. You are using chat.getTag() in a few places, maybe assign a variable to it for compactness?

By implementing these three suggestions, I got down to:

T tag = chat.getTag(); // whatever the actual Type is
return tag.equals(TAG_UPDATE_MSG) ? 
    Resources.string(UserRelationshipInfoManager.getInstance().isUserMyFriend(chat.getUserId()) ? 
        R.string.text_user_is_friend : 
        R.string.text_user_is_stranger) : 
    tag.equals(TAG_TEXT_MSG) ? 
        new TextInfo(chat).getText() : 
        String.format(Resources.string(chat.getType() == DIRECT_MESSAGE ? 
            R.string.text_new_direct_message : 
            R.string.text_new_message
        ), chat.getUserDisplayName(), Resources.string(getMessageType(tag)));

I must admit, this is taking things a little to the extreme, and I'm conveniently relying on the assumption that passing extra arguments into String.format() will be silently ignored (tested). Otherwise, I'm actually pretty OK with what you have in your question. :)

There are some Suggestions regarding code :
1. Always initialise local variable (some default value or null)

String message = null;

2. Don't use unnecessary else if you can use else if instead

if (isUpdateMessage) {...
   } else{
     boolean isDirectMessage = (chat.getType() == DIRECT_MESSAGE);
        boolean isTextMessage = (chat.getTag().equals(TAG_TEXT_MSG));

        if (isDirectMessage) {...
       } else if(...

You can use else if at very first else

if (isUpdateMessage) {...

   }      
   if (chat.getType() == DIRECT_MESSAGE) {...

   } else if(...

3. Don't create new boolean if not getting use at more than one place and remove unnecessary parenthesis ( brackets).

boolean isDirectMessage = (chat.getType() == DIRECT_MESSAGE);
                          ^... two brackets not required...^

Here boolean is not required to create as we can use condition directly in if part.

4. And always call .equals() on constant, so that if other string (chat.getTag() in this case) is null then there will not be any NullPointerException

`chat.getTag().equals(TAG_TEXT_MSG)` should be `TAG_TEXT_MSG.equals(chat.getTag())`

Please find below method code after incorporating comments :

private String getNotificationDisplayString(ChatMessage chat) {
    String message = null;// initialise local variable to null 
    // check if updated message directly and isUpdateMessage removed
    if (TAG_UPDATE_MSG.equals(chat.getTag())) {
        // for update message
        UserRelationshipInfo relationshipInfo = UserRelationshipInfoManger.getInstance().getInfo(chat.getUserId());
        boolean isUserMyFriend = (relationshipInfo != null && relationshipInfo.getSource() == SOURCE_FRIEND);
        message = Resources.string(isUserMyFriend ? R.string.text_user_is_friend : R.string.text_user_is_stranger);
    }else if (chat.getType() == DIRECT_MESSAGE) {
            // for direct message (special case)
            String messageName = Resources.string(getMessageType(chat.getTag()));
            // message = "<name> sent you a direct <image/video/etc>"
            message = String.format(Resources.string(R.string.text_new_direct_message), chat.getUserInfo().getDisplayName(), messageName);
    } else if (TAG_TEXT_MSG.equals(chat.getTag())) {
            // for text (non-direct)
            message = new TextInfo(chat).getText();         
    } else {
            // for non-text (non-direct)
            // message = "<name> sent you a message"
            message = String.format(Resources.string(R.string.text_new_message), chat.getUserInfo().getDisplayName());
    }

    return message;
}

Same as h.j.k that you don't have to store the booleans for an if.

Return directly in stead of saving into message.

Result is like this :

private String getNotificationDisplayString(ChatMessage chat) {
    if (chat.getTag().equals(TAG_UPDATE_MSG)) {
        // for update message
        UserRelationshipInfo relationshipInfo = UserRelationshipInfoManger.getInstance().getInfo(chat.getUserId());
        boolean isUserMyFriend = (relationshipInfo != null && relationshipInfo.getSource() == SOURCE_FRIEND);
        return Resources.string(isUserMyFriend ? R.string.text_user_is_friend : R.string.text_user_is_stranger);
    } else if (chat.getType() == DIRECT_MESSAGE) {
        // for direct message (special case)
        String messageName = Resources.string(getMessageType(chat.getTag()));
        // message = "<name> sent you a direct <image/video/etc>"
        return String.format(Resources.string(R.string.text_new_direct_message), chat.getUserInfo().getDisplayName(), messageName);
    } else if (chat.getTag().equals(TAG_TEXT_MSG)) {
        // for text (non-direct)
        return new TextInfo(chat).getText();
    } 
    // for non-text (non-direct)
    // message = "<name> sent you a message"
    return String.format(Resources.string(R.string.text_new_message), chat.getUserInfo().getDisplayName());
}