Roddenberry thought it gave him a 'nautical' look which worked well for his character. (At least that's what he told me when I asked him at a convention around '93. Frakes, that is. Not Roddenberry.)
Lord Garth said:
Jonathan Frakes grew a beard over hiatus in 1988 and Gene Roddenberry liked it so much that he thought he should keep it.
He said the same thing at a convention in San Diego during the summer hiatus between the second and third season. IIRC, he said that he grew it because he hated shaving (or was tired of it) but was ready to have it shaved off when they returned to production. Roddenberry saw it, liked it and said it had a "nautical look," to which Frakes joked with the audience, "What the hell is a nautical look?" But he didn't care since he got to keep the beard.
The funny thing, he mentioned, was that the producers couldn't agree on how high up the cheek it should go. At first it was about mid-cheek, but then they trimmed it, then decided they liked it the other way, and so make-up had to add fake hair until it grew back. Then, of course, it had to be trimmed again because the producers wanted it shorter. Frakes said that it took a few episodes before the beard was finally trimmed the same for the rest of the season.