Off the top of my head, here are a few possibilities:
speed of searching
Searches can be slow. It isn't obvious how much this is caused by the existing data structure and code, and how much is down to the power of the hosting machine, or the amount of simultaneous traffic it is trying to support.
In so far as it is the data structure, it seems likely that some use of (lately) standard technologies would help - given my background, a SQL database and a tuned mod_perl-based Apache webserver would be the obvious choices.
It would be nice to get paged results, eg the first 30 (or whatever) results and a button to get "the next 30".
As long as the search can be expressed as a single SQL query, this can be trivially done using a LIMIT clause.
Such a facility would be made all the more useful if the first page of results also showed how many results there would be in total: "showing results 1 to 10 of 635".
It would be useful to be able to construct more complex searches, eg to combine a search for a list of numbers with a constraint that a particular keyword must be present.
Similarly, it would be useful to be able to redo a search with modified constraints (typically to make the search more restrictive if too many results are being returned).
When a sequence is discussed, the discussion is occasionally synopsised as a one-line comment on the sequence. It might be useful to be able to cross-reference from a sequence to the full discussion that relates to it.
When a web browser isn't the most convenient thing to use (eg if you want to query the database from a mobile phone, or you want to do some high-level analysis of a large chunk of the database), it would be useful to have a mechanism and standardised interchange format for querying the database.
If I understand it correctly, Marc LeBrun has been looking at defining an XML schema for an OEIS sequence with the intention that it might be used as the interchange format for a webservice. (I'm not sure if he's also considered the format for the query itself.)
updates to the database
I know nothing about the mechanisms used to update the database, but my impression is that everything is done by Neil, essentially by hand with maybe some help from scripts, and with Neil's email client as the primary queueing mechanism.
If that is the case then it might be possible to make Neil's life easier, and the whole process less reliant on his availability, by using a more distributed mechanism - I could imagine a pool of authorised maintainers having access to the list of requests for changes or new sequences, and simple facilities to accept (and so effect) the change, or to respond to the initiator with requests for clarification or further research.
Please let me know what I've got horribly wrong in any of the above, or anything I've left out.