We will never know for sure what prompted the abolition of double jeopardy, sp. However, as you rightly say, Mrs Ming battled for years for a change but, tellingly in my view, she was not successful until long after the death of Stephen Lawrence.
Of course defendants always face the prospect of being judged by somebody who knows about their past rather than solely on the facts presented to support the current prosecution. In normal circumstances this is not a very likely prospect. However the allegations against the five in this case have remained high profile for eighteen years and there cannot be many people in the country who have not heard about the case, the victim and the suspects in some way.
But the most crucial difference here is that three of the five suspects have already been tried and acquitted. Most AB-ers will know that I have no time for robbers or murderers (or even speeding motorists!) and following proper conviction they deserve about twice what they usually get. However, I am also a strong believer in due process being applied and I am fundamentally opposed to the abolition of double jeopardy. People, however odious they may be, should not spend their lives wondering whether a second prosecution is going to launched against them for a serious matter of which they have been acquitted (unless the due process was faulty due to, say, jury nobbling or a witness committing perjury).
It may well lead to cases such as this where people who are quite likely to be guilty go free. But there are many such shortcomings to our criminal justice system and they are part of the price we all pay for a system which protects us from unfair conviction.