Massachusetts is making a renewed effort to keep its home-grown college talent. The state government has redesigned a website aimed at persuaded some of the 350,000 undergraduates and graduate school students to remain after they finish school. Charles D. Baker, the Republican gubernatorial candidate, frequently cites their exodus, attributing it to the high cost of living. Governor Deval Patrick, a Democrat seeking reelection, underscores his efforts to attract and retain jobs. The website offers access to Massachusetts companies looking to hire, as well as comprehensive internship opportunities, networking events, and messages directed to students.