There are two causes for this problem. Either the partition of the SSD is corrupted or the SSD itself is failing which is very unlikely to happen. That is why you are unable to reinstall it and it is not appearing as a destination to reinstall.
You have three options in here. First is, buy Disk Warrior (powerful disk utility application)and try to repair your SSD or recover your files with it. If it is unable to fix the SSD, then it will at least help you to recover the latest files thatare readable and save them to an external drive. Get it from here : http://www.alsoft.com/diskwarrior/
Disk Warrior is costly. If you don't want to recover any files or have backup, then you don't have to buy Disk Warrior. It is not worth it if you don't have any important files/applications.Disk Warrior will not fix it if the SSD is failing.
Second is, if you have another Mac and a Thunderbolt/Firewire cable, then you can setup Target Disk Mode and this will allow you to mount the SSD of this Mac to that like an external SSD and you will be able to back up your non backed up files --> https://support.apple.com/kb/ph10725?locale=en_US
(After backing up,you can reinstall the OS X after erasing the drive.)
Third is, you can erase the SSD and try to install a fresh version of Mac OS X in it. This will erase everything from your SSD unfortunately. Boot into the Mac OS X Utilities screen just like you did before (Command and R). In there, select Disk Utility. In Disk Utility, select your SSD in left column and erase it. Then close Disk Utility and continue installation. This is the quickest solution if you don't mind losing the files in it. You can restore from Time Machine after reinstalling the OS X.
Let me know how it goes