It means a partner for the trial: to help the subject take part in it and presumably to help ensure that there are no ill effects from taking part and to report or corroborate them if there are.
It's unfortunate wording. A trial run is an exercise to test something when it is done in earnest; a trial partner could easily mean that the partnership is under test. Given the context of a medical study or trial, that meaning is unlikely (and could be positively dangerous — the subject needs to be absolutely certain that their partner is reliable).