Blackberry had black tips to his ears. It is possible that, especially when he was a kit those tips reminded people of two blackberries.
There may also be an aspect of being tenacious and difficult to eradicate.
My thinking is less that as an individual Blackberry singularly embodies tenacity as a quality, but that there are a group of rabbits many of whose whose names are those of tough plants of farm and down land, who form a tenacious group, Hazel, Dandelion, Buckthorn, Holly.
Since leaving the warren of the snares they had become warier, shrewder, a
tenacious band who understood each other and worked together.
It is worth bearing in mind that the story originally came about as a tale told to children to keep them from getting bored on car journeys and hence may not have been subject of the comprehensive type of character design popular with some fantasy writers.
In an Ask Me Anything on Reddit seven years ago Adams said, in relation to a different question:
the key thing to note is that Watership Down was not initially intended to be a published novel. The story came about spontaneously as I told it to my children. It was never meant to be scrutinised as thoroughly as it has been.
Which may suggest that not every name carries a message, though that would not prevent it being the case that he chose the names of hardy plants for hardy rabbits with some deliberation, especially if you contrast them to a character such as Strawberry; named for a soft fruit and who is more of a voluptuary.
A question was posed about the name choices in the AMA, but alas, went unanswered.