From OED the etimology of "designate" is following mentioned as...

[f. ppl. stem of L. dēsignāre to mark out, trace out, denote by some indication, contrive, devise, appoint to an office, f. de- (de- I. 3) + signāre to mark. Some of the senses of the L. verb, having come down through It. and Fr., are expressed by design; designate is a modern formation taking up the other senses: cf. F. désigner as distinct from dessiner and obs. desseigner.]

On the other, "design" is as...

[a. F. désigner (16th c. in Rabelais, in 14th c. desinner Godef. Suppl.) ‘to denote, signifie, or shew by a marke or token, to designe, prescribe, appoint’ (Cotgr.), ad. L. dēsignāre, dissignāre to mark out, trace out, denote, designate, appoint, contrive, etc., f. de- I. 2 and dis- + signāre to mark, signum mark, sign. Cf. Pr. designar, desegnar, Sp., Pg. designar, It. disegnare (in 16th c. also dissegnare, designare, Florio). In It. the vb. had in 16th c. the senses ‘to designe, contriue, plot, purpose, intend; also to draw, paint, embroither, modle, pourtray’ (Florio); thence obs. F. desseigner ‘to designe, purpose, proiect, lay a plot’ (Cotgr.), and mod.F. dessiner, in 16th c. designer, 17th c. dessigner, to design in the artistic sense. In Eng., design combines all these senses.]