Peer-discovery protocols within P2P networks are often vulnerable: because creating network identities is essentially free, adversaries can eclipse honest nodes or partition the overlay. This threat is especially acute for blockchains, whose security depends on resilient peer connectivity. We present AetherWeave, a stake-backed peer-discovery protocol that ties network participation to deposited stake, raising the cost of large-scale attacks. We prove that, with high probability, either the honest overlay remains connected or a $(1{-}δ)$-fraction of nodes in every smaller component raise an attack-detection flag -- even against a very powerful adversary. To our knowledge, AetherWeave is the first peer-discovery protocol to simultaneously provide Sybil resistance and privacy: nodes prove they hold valid stake without revealing which deposit they own, and gossiping does not expose peer-table contents. A cryptographic commitment scheme rate-limits discovery requests per round; exceeding the limit yields a publicly verifiable misbehavior proof that triggers on-chain slashing. Beyond deposit and slashing, the protocol requires no on-chain interaction, with per-node communication scaling as $O(s\sqrt{n})$. We validate our design through a mean-field analysis with closed-form convergence bounds, extensive adversarial simulations, and an end-to-end prototype built by forking Prysm, a leading Ethereum consensus client.