Many online shops have the number 24 in them, to signify 24 hours a day – you may also be able to integrate numbers into your domain in a meaningful way to make them unique. Be careful though, because many cultures attach a lot of significance to certain numbers. Make sure that your number doesn't represent bad luck or other negative things in the country you're operating in.
My personal preference is to register the name directly with a domain name registrar rather than through my web host. I've heard stories, in the past, of less-than-reputable web hosts that registered the domain under their own name, making them the owner of the domain rather than you (although I don't know if such web hosts still exist today). Registering direct with a domain name registrar allows me to make sure that I am registered as the owner, the administrative and technical contacts. Being the owner is vital — if someone else places himself as the owner (such as your web host), he can always decide to charge you some exorbitant fee for the use of the name later, and there is little you can do. The various other contacts are less vital, but may still play important roles, depending on your registrar. For example, for some registrars, the administrative contact's approval is required before a domain name is transferred out of a web host (or at least, it used to be). If he/she cannot be contacted, the technical contact is used.
A domain name consists of one or more labels, each of which is formed from the set of ASCII letters, digits, and hyphens (a-z, A-Z, 0-9, -), but not starting or ending with a hyphen. The labels are case-insensitive; for example, 'label' is equivalent to 'Label' or 'LABEL'. In the textual representation of a domain name, the labels are separated by a full stop (period).
iPage's web hosting packages start as low as $1.99 per month and they throw in domain registration for free during the first year, which makes them a bargain for very, very lean startups or one-person operations. They also have the classic domain name search portal, and provide a tutorial on the new TLDs that helps, considering a new one seems to be appearing every second or two.