The complete custom design (not just the logo) is included in the so-called graduation. Graduation is usually considered when a site consistently receives ten questions per day (see Graduation, site closure, and a clearer outlook on the health of SE sites).
In addition to the custom design, graduation also includes various other changes (e.g. removal of the ‘beta’ part of the name, moderator elections, migration paths, community ads, and increased privilege thresholds); however, the custom design is the largest part of the workload. Therefore, the current process may be divided into two steps: Sites may be considered to graduate without a custom design first, and receive a custom design at some later time (see Design-Independent Graduation is on for early September!). (Limited design changes for graduated sites that wait for their custom design are discussed here, but nothing has been decided or even implemented yet.)
This is quite the opposite direction of the feature request that is expressed in the question. Hence, I take a pessimistic view of this request. However, I am just a moderator and do not decide such things. If a constructive feature request receives clear support and consensus from the community, we can forward the request.