From your description, it sounds like the rubber has torn along the crease. In that case, your thought of bonding a strip of flexible (but tear-resistant) material to what's left sounds good. You may have to use something a little bit thinner than a tire inner tube however
Yes, that's right - hinge broke into 3 pieces. IBM's hinge rubber material is c. 0.5mm thick, and a bike inner tube is 1.0mm thick, so I've been looking around for other sources of thin rubber.
My favourite would be butyl rubber (as used to line ornamental fish ponds), and I've sent off for some samples - unfortunately 0.75mm is the thinnest available.
There is one other source of thin black rubber - and this morning I received some samples from a company who make garments for the rubber fetish community ... I only hope I don't get placed on some 'naughty knicker' mailing list as a result of asking for samples !
They kindly sent me 6" square samples of (4D brand) black latex rubber sheet in 0.5, 0.45, 0.4 and 0.33 mm thicknesses - enough to make a hundred hinges. The two thicker sizes feel about right, and I'll now experiment with various glues. The only downside with Latex is that it's not UV resistant and it might perish from contact with natural skin oils. Still, worth experimenting with - and I'll post results in due course.