Recent experimental advances highlight electron charge qubits floating above solid neon as a promising emerging platform for quantum computing, but the physical origin of single-electron lateral trapping is still not fully understood. While prior theoretical work has mainly examined electrons above bulk solid neon, experimental systems usually feature neon layers of only $\lesssim 10nm$ thickness and non-uniformity, highlighting unresolved questions about how thickness influences electron trapping. Here we theoretically investigate the effect of finite thickness and non-uniformity of solid neon layers on electron trapping. For a 10 nm layer, the electron binding energy is enhanced threefold compared to the bulk. Exploiting this thickness dependence, we propose a nanopatterned-substrate mechanism in which engineered thickness variations generate lateral trapping potentials for electrons. The lateral trapping potential can be finely tuned by a perpendicular electric field. Such non-uniform-thickness induced electron charge qubits open a viable pathway toward building multi-qubit systems for quantum computation.