There's several items to verify when you are not able to establish a connection with your SQL Server from Remote Desktop Manager for iOS.
•Validate that the appropriate server name has been entered
•Try to connect on your SQL Server using the IP Address instead of the server name
•Make sure that you have entered the appropriate username and paMoressword to connect on the server
•If you are using Integrated Security to authenticate on your server, make sure to configure the domain in the username field on your device. Ex: Domain\Username
•Try to ping the server with the ping tool available in Remote Desktop Manager for iOS in More - Tools - Ping
•Validate that you are able to connect from Remote Desktop Manager Windows or Mac edition
The connection timeout issue can be caused by a DNS issue. As exampe, the DNS can return two IP address for the same server. A restart of the SQL instance may be needed in that case.
When a ping is performed, make sure that the ping returns the same address that the one shown in SQL Configuration Manager.