5.1 is the latest I see available from any of the usual yum repositories. If you want anything newer, you'll have to build it yourself or find a repository hosting it.

You can upgrade MySQL by installing the Remi repo:

sudo rpm -Uvh http://download.fedoraproject.org/pub/epel/6/i386/epel-release-6-7.noarch.rpm
sudo rpm -Uvh http://rpms.famillecollet.com/enterprise/remi-release-6.rpm 

and then installing the new MySQL package:

sudo yum --enablerepo=remi install mysql mysql-server

I upgraded using this tutorial on DO:
https://www.digitalocean.com/community/tutorials/how-to-install-mysql-5-6-from-official-yum-repositories

After installing, I typed in the following but MySQL wouldn't start:

service mysqld start

I even upgraded the droplet to 1 GB thinking that the newer mysql server version might require more RAM, but it doesn't work.

Any help would be appreciated. There is no sudo in the command as I had escalated to su -

@ Kamal, this is a bit late, but if you chance across this thread, here's the output of the mysqld.log file:

140731 22:11:59 mysqld_safe Starting mysqld daemon with databases from /var/lib/mysql
2014-07-31 22:11:59 0 [Warning] TIMESTAMP with implicit DEFAULT value is deprecated. Please use --explicit_defaults_for_timestamp server option (see documen$
2014-07-31 22:11:59 17699 [Note] Plugin 'FEDERATED' is disabled.
2014-07-31 22:11:59 17699 [Note] InnoDB: Using atomics to ref count buffer pool pages
2014-07-31 22:11:59 17699 [Note] InnoDB: The InnoDB memory heap is disabled
2014-07-31 22:11:59 17699 [Note] InnoDB: Mutexes and rw_locks use GCC atomic builtins
2014-07-31 22:11:59 17699 [Note] InnoDB: Memory barrier is not used
2014-07-31 22:11:59 17699 [Note] InnoDB: Compressed tables use zlib 1.2.3
2014-07-31 22:11:59 17699 [Note] InnoDB: Using Linux native AIO
2014-07-31 22:11:59 17699 [Note] InnoDB: Not using CPU crc32 instructions
2014-07-31 22:11:59 17699 [Note] InnoDB: Initializing buffer pool, size = 128.0M
InnoDB: mmap(136019968 bytes) failed; errno 12
2014-07-31 22:11:59 17699 [ERROR] InnoDB: Cannot allocate memory for the buffer pool
2014-07-31 22:11:59 17699 [ERROR] Plugin 'InnoDB' init function returned error.
2014-07-31 22:11:59 17699 [ERROR] Plugin 'InnoDB' registration as a STORAGE ENGINE failed.
2014-07-31 22:11:59 17699 [ERROR] Unknown/unsupported storage engine: InnoDB
2014-07-31 22:11:59 17699 [ERROR] Aborting

2014-07-31 22:11:59 17699 [Note] Binlog end
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'partition'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'ARCHIVE'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_DATAFILES'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_TABLESPACES'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_FOREIGN_COLS'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_FOREIGN'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_FIELDS'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_COLUMNS'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_INDEXES'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_TABLESTATS'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_SYS_TABLES'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_FT_INDEX_TABLE'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_FT_INDEX_CACHE'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_FT_CONFIG'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_FT_BEING_DELETED'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_FT_DELETED'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_FT_DEFAULT_STOPWORD'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_METRICS'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_BUFFER_POOL_STATS'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_BUFFER_PAGE_LRU'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_BUFFER_PAGE'
2014-07-31 22:11:59 17699 [Note] Shutting down plugin 'INNODB_CMP_PER_INDEX_RESET'

Any pointers on how to proceed?

Thanks in advance.