I live in Roppongi, and there are a lot of things to see and do here, day or night. For daytime activities, you can visit the National Art Center, which is holding a Renoir exhibit right now. You can enjoy Hikokicho park behind the Midtown shopping center. The park has a nice pond, stream, and waterfalls, and is a good place for a picnic. You can have lunch at Sushi Zanmai, there are two restaurants in Roppongi, one on Roppongi Street, another on Gaien Higashi Street.
At the shopping center in Roppongi Hills, you can have pizza by the garden, and on weekends there is usually some kind of event at the Arena area. This week's event was a farm display, with milk, cheese, and ice cream straight from "local" farms.
Another interesting place is Aoyama cemetery, the largest and most famous cemetery in Tokyo, and the resting place of the rich, famous, and a few shoguns.
If you walk down Higashi Gaien street, it will take you past the Russian embassy, and to Tokyo Tower.
The only bad thing about Roppongi is that it is not a good place to experience "real" Japan, or see any tradition or culture. It is a busy place with many foreign visitors, hustlers, and, at night, countless drunks. It is a great place to see while you are in Japan, just to experience the most "international" part of the country.