It's perfectly possible that your brother's client is genuinely an absent-minded professor who keeps forgetting to put the cheque in the post, but even if that is the case it's no justification for failure to pay.
Given that it's a relatively large sum and that it's eight months overdue, your brother is perfectly justified to get tough about this - it's gone beyond being reasonable.
Step one: write to the client informing him that if the debt is not payed within 14 days of the date of the letter your brother will file a claim against him. Send the letter by recorded delivery.
Step two: if payment has not been made after 14 days, go ahead and file a claim in the county court. This will cost £250, which must be paid up front but will be recoverable.
Step three: if the court finds against your brother's client it will order him to pay your brother the sum outstanding plus the £250 court fee. If he still fails to pay after a certain amount of time then your brother can ask the court to appoint bailiffs to collect the debt (this will cost an additional £50 but again will be recoverable).
Frankly I doubt it will get that far. Unless the client feels he has some genuine reason for not paying your brother, the initial letter threatening legal action will probably galvanise him into action.
Incidentally, one thing your brother might want to consider is interest. Was there any provision in your brother's contract with this client for interest on late payments? Even if there wasn't, your brother might feel that it's reasonable to claim interest. However that might cause further delays if the client objects so may not be in your brother's best interests.