Drop all the non-essential objects, such as indexes, from the DB (or a copy of it).
Some further ideas on Stack Overflow in the Q & A Reducing Size Of SQL Backup?.
There are tools which will extract a referentially correct set of data from a DB. The one I used was for DB2, however. If I find a link for SQL Server I'll edit it in.
Edit: SQL Server Data Tools (SSDT) allows creation of a complete, empty schema and the choice of data. If you are already using Visual Studio this would be especially worth considering. Redgate has a full suite of tools that would achieve this, too.
Of course installing more software will further reduce the space available for the new database.