The most effective tools I have seen used are Mailing Lists, and Event Reminders. Some site also provide separate Generic lists for events, exhibitions or shopping. Provided your website has a large development budget an investment in a Customized mailing list might also want to be made. A recent addition to several museum websites is the Event Reminder. If the web team or museum staff has questions on what kind of experience visitors to your website are looking for, the best course of action is to post a survey to the site.
Some sites present and invite to take surveys via a pop-up window.
I find this intrusive and believe that the request should be build into the home page of the site. Online only events include chats, and special shopping opportunities. Invite a curator to a chat room to answer visitor questions. If they are not willing to sit down and do this in real time, let visitors ask questions over a period of time via e-mail, present the questions to the curator and then post the answers to the site. Shopping opportunities include discounts or pre-ordering of exhibition related merchandise or online book signings.