This thesis investigates the design possibilities for the mechanical elastic suspension of the test mass of the new concept LIG-A accelerometer, on board the space mission LIG-A-CUBE, developed with the INRiM research institute. The space mission goal is to demonstrate the feasibility of laser interferometry as a technology for acceleration readings. This new concept of accelerometers for space applications grant better accuracy performance than any existing technology. To achieve such low noise outputs, the accelerometer requires a mechanical suspension of the test mass with a main resonance frequency less than 1 Hz and 6 degrees of freedom of motion. In order to achieve this goals two design solution are considered: the use of a machined helical spring or a multiple sheets elastic suspension.