When creating a SQLConnection you can give it a name with the "Name" option. The connections are also added to the output of the SQLConnections function.
OpenSQLConnection[
JDBC["HSQL(Standalone)", ToFileName[{$TemporaryDirectory, "SE Question"}, "Ex1"] ],
"Name" -> "Example1"];
OpenSQLConnection[
JDBC["HSQL(Standalone)", ToFileName[{$TemporaryDirectory, "SE Question"}, "Ex2"] ],
"Name" -> "Example2"];
SQLConnections[]

Later on I would like to select a particular connection from SQLConnections based on its name. However, SQLConnectionInformation does not have "Name" as supported property
SQLConnectionInformation[#, "Name"] & /@ SQLConnections[]
DatabaseLink`JDBC::error
{$Failed, $Failed}
and there does not seem to be a name property on the connections
Through@SQLConnections[]["Name"]

and SQLConnections does not take a name parameter
SQLConnections["Example1"]
SQLConnections["Example1"]
How do I select a connection from SQLConnections by its "Name" property?
Clean-up:
CloseSQLConnection /@ SQLConnections[];
Update
WRI Support have provided a method to get at the "Name". I have also requested that a SQLConnection object support "Name" and "Id" as properties.
"Name" is an option. Therefore,
"Name" /. Options[conn, "Name"]
will provide conn's "Name" option.
For the "Id" property WRI said it is the third part of its FullForm. Therefore,
conn[[3]]
will provide conn's "Id" property. This method seems particularly risky since there is no guarantee that this position will contain this property in future versions.
Since you can see these values in the Format form of a SQLConnection in a notebook I think it is reasonable to expect to be able to get at these values without workarounds. Particularly since there is no other means of selecting a connection out of the SQLConnections list. Fingers are optimistically crossed.