It is known that  the threshold resonance in a quantum waveguide may lead to the appearance of a near-threshold eigenvalue below the continuous spectrum while an embedded eigenvalue above the threshold can exist only in the case when the threshold implies a true eigenvalue with the eigenfunction in the Sobolev space H^1. It will be demonstrated that the effect of the threshold resonace in an elastic waveguide (described by either the Lame system, or the Kirchhoff equation) is rather different: it may produce a near-threshold eigenvalue embedded into the continuous spectrum and therefore, situated above the threshold.