Install OS X Lion on a Mac mini
1GB of RAM is insufficient for OS X 10.7 "Lion."  You'll need 2GB or more.:

OS X Lion - Technical Specifications
General requirements


Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7, or
Xeon processor 
2GB of memory 
OS X v10.6.6 or later (v10.6.8 recommended)
7GB of available space