Hannibal is seen as great for a number of reasons, however as you mentioned we have to be careful about our sources and what they tell us. With such a limited amount of sources to Appian, Livy and Polybius to name a few who were Roman.
However, while we must be careful around some of what they say, the general history wont have been altered too much as this would have been to obvious. So when Livy tells us that Hannibal when he wintered in Capua, he claims that it was as if another army had left, the once disciplined army was gone instead this discipline was replaced by relationships with prostitutes and alcohol. ^1 We must be cautious, as what we have to remember was that the first 3 major battles of Trebia, Trasimene and Cannae were all within the first 3 years of the war, Hannibal continued to plague Rome till 202 BC. If his army was as Livy suggests no longer disciplined, then he would not have been able to easily stay in Italy for the next 12 years building a foot hold in the South of Italy.
As just mentioned, it was the understanding that Hannibal had that allowed him to be such thorn for Rome. Hannibal was raised by his Father in Spain, he had been part of war his whole life, this gave him a natural base to work from. Such as his exceptional use of cavalry due to his Iberian and African roots, when we look to most battles we see the advantages Hannibal gained via use of his cavalry. Hannibal also understood he could not siege major towns for he had to keep moving to gain supplies and to stop himself being surrounded. Some ask why did Hannibal not march on Rome after Trasimene, and to that I like Beer's response: to think that he could march on Rome "betrays a complete lack of appreciation of what Hannibal’s real problem and aim were." stating it was not the annihilation of Rome that Hannibal was after but to reduce its political power so that Rome was no longer able to threaten Carthage and it’s prosperity. ^3
When Hannibal invaded Rome, their strategy was effectively to out number their opponents to win. We can see this further by the distaste many had for Fabian who understood that Hannibal could not be met in a drawn battle but had to be contained, however, he came to be named the 'delayer' as many saw him as a coward. Cottrell also states that Scipio Africanus who finally defeated Hannibal learnt his stratagems from Hannibal.^2 The matter is, is that Hannibal aided in the development of Rome, he was studied by generals and forced Rome to adapt and overcome or be crippled. Rome overcame, but this does not discredit Hannibal for his situation was dire. Hannibal alone tied down 13 legions in Italy of the 22 deployed in the year 215 BC table^4 . He was outnumbered, out supplied and isolated and surrounded. Yet he continued to pillage Italy till he was recalled to Zama. All of this considered, it is hard to see how Roman sources could exaggerate so much. Clearly Hannibal was greater then the Roman generals that he faced, when we consider his situation this only strengthens that notion.
On top of this we must consider the fact that Carthage did not invest much into the Italian campaign at all, Carthage saw this war for the most part as a war to regain what they had lost in the First Punic War, Corsica, Sardinia and Sicily and to this extent they invested more manpower to these ends then they did to Hannibal. While Rome did have a strong naval force during the Second Punic War the lack of support from Carthage is a factor to consider in evaluating Hannibal's greatness for his campaign in Italy, while the other fronts did draw some attention away from his Italy campaign, he was mostly isolated and alone.
Due to this isolation also we must consider the composition of his force, he had Gual's, Iberians, Africans and some Greek/Italian contingents from his allies. This fact alone is rather astonishing, to keep such a divided army together in the middle of enemy territory is a feat in of itself. Lazenby has described the Gual's as "dubious of loyalty" but they caused little trouble for Hannibal. Hannibal's veteran soldiers were feared till the end even in the battle of Zama, his third line were feared.^5
The numerical figures given for the conflicts are not going to be accurate of course and we must be attentive in our estimations to what the actual figures were, as our sources are Roman there are of course going to be some skew in them. However, as mentioned the basic outline we accept for the most part, and as mentioned, the basic outline indicates that Hannibal Barca was a great general even if the causality figures were exaggerated.

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^1 Livy, Hannibal's War, Books 21-20, translated by J. C. Yardley (Page 156/ Book 23 chapter 18) Link on Perseus
^2 Enemy of Rome, by Leonard Cottrell (Page 187)
^3 Hannibal: Challenging Rome’s Supremacy, by Sir Gavin de Beer. (Page197) ^4 The Campaigns of Hannibal, P. L. MacDougall (Page 133)
^5 Hannibal’s War, J.F. Lazenby (Page 57)