I often get brides whom request adding ribbon or twine to their wedding invitations. Both are a great added detail on the right invitation.
The Twine... great if you're after a more casual look. There is burlap style twine which has a craft paper colour tone and comes is different thicknesses. The baker's twine is also very popular and comes in many colours.
The Ribbon... has a certain elegance and comes in many colours and sizes. For an invitation I would suggest going with the thinest ribbon possible. Ribbons sometime come in a string looking format or in 1/8" wide. I recommend this because a ribbon will get bunched up inside the outer envelope and may end up looking wrinkled when pulled out of the envelope. Thick ribbons will also create a big knot and add a certain thickness to your outer envelopes. Keep in mind that thickness can cause your final postage to go up.