Once you've chosen a museum, call to find out who handles the type of item you want to donate. Some museums want you to pitch your heirloom in a letter; others will talk with you on the phone.
Describe the item, its history, and why you think it belongs in the museum. If the curator is interested, he or she will want to see the heirloom and possibly show it to an acquisition committee. The museum may take temporary custody of the item while it makes a decision. Now, you wait to hear from the curator or director.