I'd probably go with a joint role - System & Network Administrator and Solution Developer - covers both the design stuff that directly contributes to the unique needs of the business as well as the general administration stuff like staging servers and documenting the network.
Conversely, on keeping it simple - Senior Engineer works too. Nice and simple, fits most positions that you would otherwise qualify for in a bigger company with more specialization. Vague enough that anyone reading your resume would have to look at the details to see what it means, which is fine, since you do more diverse work than can easily be summed up in a title.
A general rule of thumb I see:
Administrator - for keeping things running in some operational context. Also a strong bent towards solving problems in a running system that users depend on. Descriptor before "Admin" is what speciality (Network, System, Web Services, etc.)
Engineer - may administrate, may create new systems, very open ended.
Developer - makes stuff from raw materials to fit a business need. More than installing COTS products, involves some level of customization and some work translating from what the business people want to what the system actually does. Some knowledge of requirements, design, implementation, testing.
Then in rank:
- No prefix - probably 1-5 years, may need oversight
- Senior - capable of digesting a moderately hard problem, does not require oversight, only needs management backing for really tough stuff. Fully capable with any stakeholder involvement that is standard for the role.
- Principle - a go-to guru for something, able to not only learn unknown material very quickly, but can impart it at all levels of the organization. Almost never needs management involvement, probably relates to management at a peer-to-peer level.
There's a few titles around that imply some level of leadership:
Lead - gets other people mobilized in some way towards solving a technical problem. Some level of technical mentorship responsibility, some of the work requires digesting and breaking down and taking responsibility for other people's work.
Manager - a bigger focus than a lead, but also has some level of human development and legal accountability - does personnel reviews, contributes to input for objectives and possibly raises/incentives/bonuses. The big difference being a "manager" probably holds some legal accountability in the company's name and so is more aware of and sensitive to the legal risks (sexual harassment, fair hiring, etc)
If you don't take accountability in any way for other people, because you are entirely a one-man shop, I'd avoid these titles.
A little fun doesn't hurt. I've heard plenty of fun titles around, and it looks like there are plenty in the comments. If you have a fun company and want to show a little spunk, go for it, but have a backup in mind for formal situations and for helping people figure out what to do with you in less one-on-one situations. For example, conferences, resumes, and other application things - a person sorting resumes is going to say "Superman?? Whatever..." and just shove you to whatever pile appeals to them. Chances are good they have a "Admin" track and a "Developer" track, so although you're probably interested in both, at least you had a chance of getting shuffled in the right direction.
So get the fun name, stick it on business cards and use it to give yourself joy when answering the phone - but find some way of having a serious title when you need it.