Graphene, carbon nanotubes (CNT), and carbon nanofibers (CNF) are the most studied nanocarbonaceous fillers for polymer-based composite fabrication due to their excellent overall properties. The combination of thermoplastic elastomers with excellent mechanical properties (e.g., styrene-b-(ethylene-co-butylene)-b-styrene (SEBS)) and conductive nanofillers such as those mentioned previously opens the way to the preparation of multifunctional materials for large-strain (up to 10% or even above) sensor applications. This work reports on the influence of different nanofillers (CNT, CNF, and graphene) on the properties of a SEBS matrix. It is shown that the overall properties of the composites depend on filler type and content, with special influence on the electrical properties. CNT/SEBS composites presented a percolation threshold near 1 wt.% filler content, whereas CNF and graphene-based composites showed a percolation threshold above 5 wt.%. Maximum strain remained similar for most filler types and contents, except for the largest filler contents (1 wt.% or more) in graphene (G)/SEBS composites, showing a reduction from 600% for SEBS to 150% for 5G/SEBS. Electromechanical properties of CNT/SEBS composite for strains up to 10% showed a gauge factor (GF) varying from 2 to 2.5 for different applied strains. The electrical conductivity of the G and CNF composites at up to 5 wt.% filler content was not suitable for the development of piezoresistive sensing materials. We performed thermal ageing at 120 °C for 1, 24, and 72 h for SEBS and its composites with 5 wt.% nanofiller content in order to evaluate the stability of the material properties for high-temperature applications. The mechanical, thermal, and chemical properties of SEBS and the composites were identical to those of pristine composites, but the electrical conductivity decreased by near one order of magnitude and the GF decreased to values between 0.5 and 1 in aged CNT/SEBS composites. Thus, the materials can still be used as large-deformation sensors, but the reduction of both electrical and electromechanical response has to be considered.