If someone dies then they are essentially entitled to the full personal allowance for that tax year, regardless of the point in time when they died. So, for example, if someone dies on 7th April 2012, they will be entitled to the full personal tax allowance for 2012/13.
So it is likely that the earlier in the tax year that someone dies, the more likely there is to be a (bigger) tax rebate due, as the tax paid to date would assume that tax was being paid over the whole of the tax year.
You will need to notify the Inland Revenue that the person has died, and if they normally completed a tax return themselves then you will need to complete one for their financial affairs up to the date of their death (and you will have to pay any tax due before you can distribute any assets).
You can find out more information about the tax situation when someone has died by checking on the Inland Revenue website.