Typo, fixed now. \nabla^s is the symmetric gradient (the strain rate operator). \sigma = 2\mu \nabla^s u - p I is how the stress tensor is defined for an incompressible newtonian fluid.
This is not related to the normal component boundary condition though… As you indicated in your initial post, the Dirichlet condition for the pressure problem occurs at the Neumann (or really, the natural) boundaries for the Navier-Stokes problem. For the Navier-Stokes equations, the natural boundary condition is:
\boldsymbol{\sigma} \cdot \boldsymbol{n} = \boldsymbol{t}_N
i.e.,
2\mu \nabla^s \boldsymbol{u} \cdot \boldsymbol{n} - p \boldsymbol{n} = \boldsymbol{t}_N
Or, by dotting with \boldsymbol{n}
p = (2\mu \nabla^s \boldsymbol{u} \cdot \boldsymbol{n}) \cdot \boldsymbol{n} - \boldsymbol{t} \cdot \boldsymbol{n}
Which gives you your Dirichlet expression for p. That Dirichlet expression fixes the constant you’re referring to.