American palaeontologist Henry Osborn first described T. rex but initially expressed doubts that the diminutive arms he found belonged to this enormous animal. After investigating further and finding it to be true, he considered their purpose and advanced the first theory in 1906: that they were used as ‘graspers’ or stabilizers during copulation.
But this was only the start of a debate that still rages. In 1970 British palaeontologist Barney Newman suggested that the small arms braced the beast's body as it stood up.
Another, contrasting theory suggested that the small limbs are vestigials (degenerated organs that have lost their use).
To add to the confusion it seems that the muscular of the T. rex was very well defined and though the arms were disproportional to the animal's body, they were still very strong. So some people consider that the most useful function of the limbs was to be used as meat hooks while the animal fed. This theory is supported by the beautifully sharp claws on T. rex’s two fingers.
In contrast, Greg Erickson has recently commented on a biomechanical analyses of a new specimen. The aim of this study was to try and understand the physical capacities of T. rex's diminutive structures. Erickson has concluded that the elbow could not be extended beyond 90° and though the arms were very strong (they could lift about 180 kg or 400lbs) they had a very limited side-to-side and up-and-down motion.
It is also known that T. rex's arms were often broken (and mended) during its life, suggesting that the arms were poorly suited for their function that these animals could go without using their arms for extended periods of time.
I read an interesting theory a while back that suggested the arms were most useful in juveniles, as a counterbalance to the young dinsaurs fast quickly. The suggestion was made that the arms were proportionally longer in childhood compared to adulthood. However, as far as I know this theory did not gain popularity.
The truth is that no one knows the answer. Scientific inquiry into this issue has been going on for a century and it doesn’t seem likely that the mystery will be solved soon. If you have any other ideas, please let me know!