By definition, a self-signed certificate can be trusted only through direct trust, i.e. what Web browsers like Firefox show as the "allow exception" process. One very specific certificate, down to the last bit, is declared as "trusted". Nothing can be changed in a certificate without exiting from this model, and, in particular, the expiry date, which is part of the data contained in the certificate.
You can imagine renewal as a kind of family thing: when a certificate is "renewed", it is actually replaced by a younger sibling. Clients accept the new certificate silently because it shares the same ancestry as the previous certificate. Self-signed certificates are intrinsic orphans: they have no ancestry. Hence, no sibling, and no automatic transmission.
(Apart from this ancestry thing, renewal is the creation of a new certificate. Certificates are immutable. "Renewal" is a way of thinking about the relationship between the old and the new certificates.)
If you want to be able to do silent renewals, then you need a self-signed CA certificate. You emit certificates for your server(s) from that CA, and you ask your clients to trust that CA. Of course this is asking a lot: a CA that you trust is a CA that can fake the whole Internet in your eyes. Basically, this solution is about creating and maintaining your own CA, which is a responsibility and some work.
Next time you produce a self-signed certificate, make it long-lived. Certificates expire mostly in order to make revocation work (certificate expiry prevents CRL from growing indefinitely). For a self-signed certificate, there is no revocation, so you can make the certificate valid for 20 years. Or for 2000 years, for that matter (although the Year 2038 Problem might show up at some point, depending on the client software).
Best Answer
Yes. You need to import the certificate into Chrome (after exporting it to a file, if you have not done that yet). You can import certificates in the certificate dialog, which you can reach via Settings / Advanced / Manage Certificates.
Some caveats:
For details, see for example this question on StackOverflow:
Getting Chrome to accept self-signed localhost certificate