Unfortunately for you, the short answer is no.

The longer answer: Yes, the Roman legions did have names or nicknames. Some had quite cool stories behind their names, such as Caesar's Legio X Equestris, i.e. "The tenth mounted legion"so called because at some point Caesar went to a meeting where he was only allowed to bring a few cavalrymen for escort, and he wouldn't entrust his escort to anyone but his legionaries and so he mounted them up. (The legionaries joking he was raising them to Equestrian/"knightly" status, i.e. the second-highest rank of Roman nobility. The legion itself wasn't actually mounted, of course, and the name was more of a nickname.)

Some had quite impressive and cool-sounding names, such as Legio XX Valeria Victrix, "20th Legion, "valiant and victorious" though more literally translated as "victorious black eagle." They got their name (probably) for their part in defeating Boudicca. Or Legio VI Ferrata, "The 6th Ironclad legion," or perhaps "the ironsides," later earning the further names Fidelia Constans, "loyal and steadfast." Or how about Legio XII Fulminata, "12th legion, the lightning-throwers."

The problem for you is that all these names and formations, and indeed the very idea of a permanent, long-term Roman military unit with a unit identity and history, is one that only started in the time of the Principate. The legions I mentioned date back to the very late republic at the earliest, or otherwise were raised by Caesar or Augustus.

In the time of the Punic Wars, the Roman army was still very much a citizen-levy or part-time soldiers, joining the ranks out of loyalty to their city or to seek glory and loot. (And probably both.) Legions were raised every year, but also renumbered every year, and when a war was over the soldiers would go back home to their farms and become civilians again.

They did have standards, though: 5 per legion, representing an eagle, a horse, a bull, a wolf and a boar. (Later, all these would be replaced by the silver eagle as the principal standard.)

The second Punic War was a conflict unprecedented in its scale and scope, and saw many more legions raised than the Romans would for generations to come, and proportionally represented the greatest investment of man-power they ever made, but the formations were still temporary and did not carry famous names.

That said, there is one possible exception: the so-called Cannae legions. But I doubt these are exactly the type of inspirational stories you're looking for.

As you are probably aware, the Romans suffered one of the most devastating defeats of their history at the battle of Cannae in 216 B.C. The defeat, however, was not total, and the survivors were re-constituted in 2 legions. However, the Romans blamed them for the loss, and exiled them from Sicily and forbade them to put down their arms. They garrisoned and fought on that island for over a decade.

However, redemption would be theirs: finally, in the year 205 B.C., the young consul Scipio was to take the fight to the enemy. Scipio had been instrumental in turning the tide of the war in Spain after his uncle and father had been killed there by the army of Hannibal's brother Hasdrubal. Now, despite some doubt among his fellow Roman noblemen, Scipio's plan to invade Africa was enacted, and the Cannae legions would form the heart of his invasion force. (Reinforced by a large contingent of volunteers, since after more than a decade of continuous service, there can't have been more than half of them left.) With this force, Scipio crossed the Mediterranean and eventually defeated Hannibal at the battle of Zama, ending the war in Rome's favour. (Thanks in no small part to the support of Massinissa and the Numidians, I might add, but that is neither here nor there.)

It's quite a famous story, but I'm not sure it fits on a tatoo. :-)

Sources:

Goldsworthy, The Complete Roman Army and The Fall of Carthage

www.Livius.org is a great resource by a Dutch historian with an alphabetical and chronological list of all known legions and their histories. It also shows symbols where relevant, such as the leaping boar for Valeria Victrix. Should help you out.