I've worked for a group called "Repair Cafe" on and off, and I can assure you that #1 isn't really an option.  Unfortunately, the world has moved to everything being disposable - although laptops are at least disposable in smaller chunks.  A repair shop would simply replace it.  You can almost certainly do it yourself - most optical drives come out after removing a single screw on the back (the location isn't obvious - consult YouTube for your model) and a gentle pull.  If the machine is relatively modern, the drive is probably SATA and that means the connector and even the shape of the drive is semi-generic, so replacing it should be quite easy - Canada Computers was carrying the replacement drives a year and a half ago for ~$20 (but check the return policy first).  It may not sit perfectly or exactly match the colour, but it should work fine.  If it's not SATA, you'll have to get a machine-specific replacement - and those are harder to find and more expensive.

Even if it's SATA, I would agree with Lennart's suggestion to get an external - unless, as someone else said, you have to carry it a lot.  He's right: hardly any laptops have optical drives anymore, and this will future-proof you since you probably have a fair number of DVDs around.  Someone else suggested BluRay - I don't think you can play DRM BluRay media under Linux, but if it was a writer you could use it as a backup (not my first choice with external HDs so cheap, but definitely an option, and possibly useful if you shoot your own videos).

Buying the external also offers another option hinted at by Lennart: the dead drive can be replaced by a hard drive holder.  I did this in one old laptop because it had a dead 1.8" HD (yup, 1.8" - you know how hard those are to replace?!).  So I removed the optical drive and replaced it with a device the same shape and size that now holds a SATA 2.5" drive that acts as the primary drive.  So you could add a secondary HD if that appeals.