MS SQL Server using SSL Connection over the Web?
According to Books Online, the certificate used for encrypting connections must meet the following requirements:


The certificate must be issued for Server Authentication
The name (common name or CN) on the certificate must match the full-qualified domain name used by clients to connect.


If those two conditions are not met, the certificate will not show in the drop-down listing. Additionally, the service account for SQL Server must have read access to the private key of the certificate. Without that right, you will be able to select the certificate, but the service will not start.

Since this is a test system you intend to use as proof-of-concept, a self-signed certificate should be adequate. I am not sure what language you are using for your client application, but this link provides options available for connection strings in .Net. You should review at least the following two options.


Encrypt - Tells the client application to force encryption.
TrustServerCertificate - Tells the client application to skip validating the certificate chain. This is important with a self-signed certificate as the chain likely will not be trusted.