Topolectrical circuits exploit the equivalence between circuit Laplacians and tight-binding Hamiltonians, offering a highly controllable platform for studying band-topological phenomena. Previous investigations have mainly focused on periodic lattice networks, where topological invariants are defined in momentum space and are typically revealed through boundary states. Programmable circuits with continuously tunable parameters provide a complementary route, in which external control variables define a synthetic parameter space and the geometric properties of eigenstates can be probed along closed trajectories. However, despite the mature theoretical framework of Berry phases, their quantitative extraction and active manipulation in programmable topolectrical circuits have remained experimentally underdeveloped. Here, we implement a three-node programmable topolectrical circuit that realizes a real-symmetric three-level effective Hamiltonian with continuously tunable hopping amplitudes and onsite potential using varactor-biased LC branches. By varying two normalized coupling parameters, we construct a two-dimensional synthetic parameter space in which the eigenspectrum exhibits conical intersections whose locations are determined by the onsite potential. At each parameter point, eigenmodes are reconstructed from impedance-matrix measurements, and the Berry phase is evaluated through a discrete Wilson loop along closed paths. For a loop enclosing a conical intersection between the two lowest bands, we obtain a normalized Wilson-loop phase of |\gamma_\rmWL|/\pi=0.994, in excellent agreement with the theoretical value of π. In contrast, a loop that encloses no relevant degeneracy yields |\gamma_\rmWL|/\pi=0.004, consistent with a trivial Berry phase. This quantized response remains stable over a frequency window of 332–340 kHz around the design frequency. Furthermore, by tuning the relative onsite potential \varepsilon_a of a single node while keeping the measurement path fixed, we controllably displace the conical intersection in synthetic parameter space. An analytical critical value \varepsilon_a^*=-3/2 is obtained, at which the degeneracy crosses the path boundary. Experimentally, setting \varepsilon_a to 0 and –2.2 produces Wilson-loop phases of 0.994\pi and 0.004\pi, respectively, directly verifying the predicted discrete Berry-phase transition between π and 0. These results establish programmable topolectrical circuits as a quantitative platform for measuring and controlling eigenstate geometric phases in synthetic parameter space, with potential extensions to higher-dimensional parameter spaces and non-Hermitian regimes.