I would go this way
First please!.... Name the model SAME as the table name.
I would not create username field again in blogs table.
So...

Blogs Table (Blog model)

Schema::create('blogs', function(Blueprint $table)
{
    $table->increments('id');
    $table->integer('user_id')->unsigned()->index();
    $table->foreign('user_id')->references('id')->on('users')->onDelete('cascade');
    $table->string('title');
    $table->longtext('message');
    $table->timestamps();
});

In your User model

public function blog()
{
   return $this->hasMany('Blog')
}

In your Blog model

public function user()
{
   return $this->belongsTo('User')
}

Now the Blog is in relation to User, so whenever User change the name, you can access the username for the blog by

{{ $blog->user->username}}

@Torchsk, It's not about fetching bro. I am asking about update, ? i.e, if i update user table, will it auto update blog table with modified username. ?

No it won't. As TorchSK mentioned, you should use the users ID rather than their username. From what you have said, a user seems to be able to change their username, where as the users ID will never change, so use that to refer to a user in other tables.

Well yeah.. i was bit in a hurry so I did not explain myself...
But as shabushabu said, it is better to reference to user ID instead of username, while ID never change...

If there is not a special reason why to have username attribute also on Blog, it is better no to.

You can always use "User's username" as "Blog's username" as each Blog belongs to user.