Updated to Drupal 7.51 and website is now stuck in maintenance mode
I am compiling here some points which I found. Following list I have ordered them from common to complex solution.

1. Clear cache:

First thing you can do is clear the cache. If you read the forum discussion on the Drupal forum page titled: Site stuck in maintenance mode!, you'll find various people solved this issue just by clearing the cache. 


Through Admin UI: As per this comment just clear the cache by going at Administration/configuration/performance. 
Using Drush: If you are using drush, you can do this by command drush cc all.
Other methods: If above both methods are not accessible for you, you can try other methods witch are listed in this article: [Drupal]Methods to clear Drupal caches


2. Login to site:

You can try some methods to login into site while still site in maintenance mode. 


Login to site using /user/login: On this question of DA: How can get my site out of maintenance mode? @Dmitry has stated in his answer,


  Just go to /user/login, it works even in maintenance mode, and then
  sign in as an admin.

On other Drupal forum QA also if you see the comment on this Link, it talks about using /user/login. I have pasted here,


  I just had a similar issue with a D7 site I was performing maintenance
  on. Logging in via http://example.com/user and
  http://example.com/?q=user wasn't working. What did, though, was
  http://example.com/user/login.



3. Changing Maintenance Mode setting:

@mixerowsky has very well explained various ways of changing the maintenance mode in his answer at this DA question: How can get my site out of maintenance mode?