Thank you all for your answers. The consensus seems to be with old age; not surprisingly. I’ll briefly explain where I’m coming from though.
Achilles choice affected others. In ancient society there was no social welfare system. Much of the family, and many servants, relied on the warrior to bring home treasures from war to retain their affluence. A servant in a good house was often treated better than a small business owner. There was no middle class, so without wealth and/or a famous name, a family would have a constant struggle to survive.
If he died an old man Achilles would have made no great contribution to the Greeks. Although by today’s standards the Battle of Troy was not over an issue worth a ten year war and thousands of deaths, in the culture of the time an insult to a king as powerful as Menelaus had to be punished, or surrounding nations would see them as being weak and quickly work to bring down the kingdom. Once this challenge was thrown down, again it could not be retracted or surrounding kingdoms would quickly attack the depleted army. The Greek army was on the edge of slaughter, and Achilles contribution to the battle essentially maintained peace in his homeland.
The rest really depends on what you want from life. @cwilbur had a good point, that we wouldn’t have heard of him if he chose the other option, but the significance of that depends on whether or not you want to be remembered.
For the record I am not sure what I would choose, there are too many angles to consider. I am quite sure I would never be offered this choice though, seeing as I would make a hopeless warrior.