If I have a good claim, why should I offer to accept a lower amount in settlement?

There are a number of reasons why making an offer a settlement is desirable, but the two main reasons are to protect your position in respect of your costs in any court proceedings, and in the hope that a settlement can be achieved, thereby avoiding the costs, time and stress of the proceedings.
There is a litigation risk in any proceedings, because ultimately it is down to the judge to decide based on all of the evidence presented to him during the trial. Therefore, no matter how strong your case is, there is always a risk that you may not succeed in the proceedings and that you could be responsible for paying the other side’s costs, in addition to your own costs.
If you are successful, the general rule for a claim of over £10,000 is that the unsuccessful party pays the successful party’s costs. However, the court will assess the amount of costs that should be paid. A general rule of thumb is that you will recover about 2/3 of your actual legal costs.
If a settlement can be reached, this avoids the uncertainty of whether you will be successful, and avoids the time, stress and on going costs of the proceedings, as well as the loss of the element of your costs which will not be recovered even if you do succeed.
Even if the settlement offer is not accepted, if you go on to a full trial and are ultimately awarded a higher sum than you offered, you have a strong argument that all of your costs should be paid in full from the time the offer was made, because the other party could and should have accepted the offer at that early stage and avoided those costs being incurred. If the offer is made pursuant to Part 36 of the Civil Procedure Rules 1998, the award of costs will be even more favourable.