A claim founded on injury to the menisci of the knee is absolutely something you should involve a solicitor for - you may end up looking at an award of damages in the region of £8k to £12k depending on severity and prognosis. It is a very serious injury indeed. Accordingly, the claim will fall to either the fast track or the multi-track, and solicitors costs will be recoverable.
Any reputable personal injury solicitor would be happy to take this sort of case on - it sounds a strong one from what you have said thus far. Do a Google search for Ward v Tesco Stores Ltd (1976) for the relevant authority on this type of claim.
As has been mentioned many solicitors will want to take a success fee of up to 25% of damages from you. There are however firms out there that do not charge a success fee, thus entitling your relative to keep all damages.