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As Olybrius' wiki notes, he was the grandson and great-grandson of emperors and a member of the Anicia bloodline. Perhaps this was done for the same reason Olybrius was married to his niece:
Anastasius wanted to strengthen his rule through a bond with the House of Theodosius through this marriage.
(Furthermore, in 512, the citizens of Constantinople openly wanted Olybrius' father, Aerobindus, to be their emperor (thanks to Anastasius' religious muddling). This goes to show what a distinguished family he belonged to.)
Considering that consuls are appointed for a one-year term, presumably it was easier to control a ten-year old than an adult in the first year of Anastasius' reign. It is telling that no second consul was appointed and that Anastasius himself became consul the next year.
(Incidentally, his grandfather, Ancius Olybrius, was also chosen as a puppet ruler. This might also have been a favourable trait.)
As for whether this was a custom or a one-off, it was very likely the latter. As mentioned above, Anastasius himself was consul the next year and, at the age of 62, he was no spring chicken. Looking at the biographies of other consuls of the time cements this view.
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By this time, consul is more a status title than any position giving political authority. So it likely is just that the emperor wanted to reward that family with a plum and for whatever reason this child was selected rather than someone more mature. Perhaps the father had already been consul.