The single most important fact to keep in mind when reviewing a fund is that there is no single most important fact.
Left tail risk in a fund investment exists for a huge number of reasons. This could range from back office compliance, risk management/derivative use policies to the possibility that the strategies they're running are negatively skewed which attracts capital but will eventually go bust.
To review the potential upside of a fund is similarly complex. Perhaps for this question, there is one single most important fact to keep in mind: that all your efforts to predict their performance are going to be almost completely futile, and in population terms you may be adding only marginally to your portfolio expected return, or you may be adding nothing. More plausible is that you are decreasing standard deviation through a review and thinking on the portfolio level, but not increasing expected return in a meaningful way. The exception would be if you are in the fund of an elite school and you get first access to the best PE deals from your alumni.
Edit: If the answer he/she was looking for was "risk management", then I accept this as a sensible question. Behavioural economics teaches us that people's utility functions are strongly loss averse, and industry knowledge tells us that fund returns are almost all the outcome of chance/luck, so risk management can in my mind be argued as the most important fact to keep in mind.