The gear linkage top-hat bushes are often a cause of side-to-side gear stick sloppiness.. I'd start with them as it's a much cheaper/easier job than changing out the stick or selector cables!
He said he'd checked the gearbox end and it wasn't that, so if we take him/her at their word then the options are fix the gear lever box or replace it. As Ed says it's a pain in the arse to get them out requiring the removal of the exhaust centre section and the heat shield above it before starting on the lever housing it's self but it isn't actually difficult. It comes out with the cables attached as I find they are far easier to release from under the battery box than at the gear lever end: Though when it comes to refitting they can be a pain to thread back through again. If there was a way to fix the slack in the pivot you would have to get it out of the car and by the time you've done all that, putting in a replacement may well seem like the more attractive option.
I've tried to attach some pictures of an old one of mine. In this picture one cable ( the fore/aft movement one) has been removed to use as a spare in another car) they are a bit blurry I'm afraid sorry; the mechanisms are very similar in all the cars. The gear stick passes through a nylon ball and socket mechanism. This is an encapsulated beast and there is no means of adjustment. the ball has some grooves in it designed to allow crap that falls into the mechanism to hide and not grind down the bearing surface, you can take off the gaiter and blow it out with an air line occasionally however when the top hat bushes on the gearbox that The Ed mentioned start to bind, a lot of owners think the stiffness is from the gear-lever end and squirt a bit of silicone lubricant down this ball joint turning the crap in the grooves into an abrasive paste, making it much more difficult to blow out, not helping the original problem and accelerating the wear. The side to side movement is effected by the side pivot mechanism ( the annodised brass finish arm). This depends on three bearings the rear cylinder in which the cylinder holding the cable arm pivots, the side pivot which moves the cable arm up and down and the same nylon ball and socket joint from which the side pivot protrudes. I doubt any of them are obtainable separately and there is no means of adjustment to them as you can probably tell from the photos You could make a bearing of the rear type from a bit of phosphor bronze in a lathe but you'd have to bore out the old nylon square and it may not even be that that's the problem so. .NO ONE and I mean NO ONE likes to to stick a couple of good British Archers fingers up to the " sorry you have to buy the whole assembly for a ridiculous price" squad than me but on this occasion sadly it looks like replacement is your best option.
