A possible source of "second signal" to a B-cell bound by specific antigen include -

Role of T cell in antibody formation: Antigen with class II MHC on antigen presenting cells activates the CD4 helper T cells. CD4 helper T cells produce cytokines that stimulate B cells which have already exposed to similar antigen. Though antibody is produced by B cells (after differentiating into plasma cells), coordination of helper (CD 4) T cells is also necessary. In antibody formation, T cells are involved earlier than B cells. After antigen exposure, antigen presenting cells present antigen to CD 4 helper T cells, which then secrete cytokines that induce differentiation of B cells ( which have already been exposed to the same antigen) to plasma cells. Plasma cells form an antibody against that specific antigen.