The answer to this question, of course depends on what one means by 'good'. I will choose a low bar for this qualification - let 'good' mean 'reasonably competent'. By reasonably competent I mean capable of obtaining tenure, funding, publications, students, etc. - or equivalent. The main virtue of this identification is that it allows me to at once claim to be a good theoretical physicist! It also makes it natural to state that one can hope to become a good theoretical physicist without being a genius. A maximalist counterexample to the approach just taken would perhaps be posed by the Landau minimum, a famously exacting set of criteria the great scientist used for selecting his students, many of whom became justly well known later.
Mathematical ability, abstract reasoning skills and some deep familiarity with physics seem to be the three requirements for becoming a good theoretical physicist. (I have heard that it is also required to have a good working knowledge of the Greek and Latin alphabets - so as not to run out of indices on vectors, or more generally, tensors!*). These skills can be learned and developed during our formal education - but see what Nobel Laureate Gerard't Hooft has to say about learning theoretical physics without going to school for it.
I think it is important to note that there are many species of theorists. They occupy various niches in the physics ecosystem. And they are all are necessary for the success of the enterprise, in my opinion. There are theorists who do only theory and no experiments (surprise!). They can be found, for example, in high-energy physics, and in - most of - condensed matter physics. But there are theorists who also do experiments, in areas such as quantum optics (that is indeed one of the attractions of the field, that the same person can do both experiment and theory). There are theorists whose work does not have immediate experimental implications (string theorists); while others are more strongly coupled to laboratory science (high energy phenomenologists). There are pen-and-paper theorists; then there are those who use sophisticated computational techniques. And so on.
For myself and my students, I try to engineer a mix of possibilities. I think doing theory that has been verified by experiment shows that the theorist is able to relate to the real world. I personally - perhaps due to the fact that I have been trained as an experimentalist - naturally interpret and suggest experiments and speak the language of experimentalists. Some of the most satisfying and impactful science I have done has emerged from such engagement with experiment. In fact I was told - not sure how true it is - in my early days as a professor that no funding agency would take me seriously unless I collaborated with an experimentalist. For the record, I have been funded both collaboratively as well as individually.
Having only experimental collaborations on a theoretical resume, however, can lead to the charge of being a mere 'calculator' - someone who is always directly following experiment, and lacks original and independent theoretical ideas of her/his own.
Doing work independently of specific experiments shows understanding of the theoretical aspects of the relevant physics. In case the work involves proposing an experiment, it leads, rather than follows, existing experiments. Increasingly more independent - from experiment - is work that introduces a new theoretical technique (e.g. density functional theory) or paradigm (e.g. Landau theory, renormalization group). Just to remind, the examples in the parentheses are all Nobel-prize winning achievements, but more mundane instances can also be substituted for them.
On the other hand, having only 'pure' theory on the resume can lead to an accusation of being a fantasizer, someone who believes in equations rather than in phenomena that can be experimentally observed - though Steven Weinberg famously warned us that the problem is not that we do not take our theories seriously but that we do not take them seriously enough.
In my opinion having both accomplishments - standalone theory as well as that coupled to experiment - brings balance to a theorist's oeuvre. Likewise, a combination of analytic and numerical techniques shows versatility. Finally, some single author papers amongst a slew of collaborative work gives an impression of intellectual independence.
I would perhaps add that in my experience being a good theoretical physicist requires an appropriate mixture of faith and doubt. If faith dominates, i.e. if one believes every idea that comes to mind, a lot of time will be wasted. This kind of attitude is warned away by sayings in the business such as 'most ideas in theoretical physics are wrong' or that 'a good exercise for a young theoretical physicist is to give up one pet idea before breakfast every day'. On the other hand, if doubt rules everything, too much skepticism will make progress impossible. Finding the right combination, of course, is an art.
* It's a coincidence that a week after writing this I am visiting the University of Crete and living the experience in reverse: because of my exposure to the Greek alphabet because I am a physicist, I can read all the signage in Greece without much trouble. Helps to pick out the flights, etc., though Google translate has to help with the rest.
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