Altering branding is generally handled in one of three ways...
- "Formerly XXXXXXX"
- Completely unrelated branding to appear as a new company
- "A subsidiary of XXXXX"
Given the clarifying comment, it's imperative to consider how you want to move forward and what you want your business to look like 1, 3, 5, + years down the road.
If the goal is to inevitably alter the brand name entirely
...then I would suggest leaving the old domain, with a redirect to the new domain. This will ensure all your existing visitors get forwarded to the primary web site. You might also consider an "I see you came from our old site. Don't forget to update bookmarks." type of message. Eventually traffic to the old domain will slow and die. At which point you can let it go. Although leaving a redirect in place is never really a bad thing.
Then you spend your time and energy on the new domain. Pretty much ignoring the old domain.
The brand should reflect, for a period of time, the previous brand name or identity. Hence the "formerly XXXX" method. This allows existing clientele to be aware that it's the same company just a new name/brand and eliminates much of the mistrust which may otherwise surface.
If the goal is to merely operate under both brands
...then you want both sites to appear completely different and never duplicate content. You're probably aware, but duplicate content on multiple web sites will lower SEO across them all. And in this case you don't necessarily need to do anything to the old site except perhaps add an advertisement for the new site/brand.
You then build out the new brand/site as a completely separate entity.
If the goal is to operate the new domain as a subset of the old domain
....then branding for the new site is often the same branding as the old site with a touch of flair - the "A subsidiary of XXXX".
FedEx does this:
Same general brand with some alterations for subsets of the company. Without being a big, internationally known company, you need to stay closer to your primary brand to remain identifiable, but the general idea is a tag line or some other unique aspect added to the existing brand t indicate a subset within the company.
You would brand the new site with this derivative brand and build it with original content related to those special services or the thought behind the subsidiary. Again, you do not want to duplicate content from the main site if SEO is a concern.
As for your girlfriend and her business....
...be careful. Realize I have no idea about your relationship. But mixing business with someone you are in a non-binding relationship with can be horrific. If you have a falling out and she is intertwined with your business ventures, a vindictive person can sabotage the business of the other. Again, I do not know either of you. But until you are married and both legally responsible for the aspects of the business. I would never, not ever, mix my business with a "girlfriend".
By mixing the domains and businesses you are both putting yourselves in a position to be responsible for the other's mistakes. If either of you were to happen to be sued for something and the businesses are mixed... there's no telling if a judge would see you as one business rather than two separate entities, making you both equally and severally liable. Same holds true for taxes. No clue where you are located, but if one of you fails to pay taxes, the IRS might see you as one business and go after the other for payment. All may seem wonderful and dandy now... but relationships can change rapidly often resulting if severe regrets for commingling things, especially things centered around income.
Have her purchase her own domain. You can support her by building the site or with the technical aspects or anything else she may need, even pay for her domain if you feel you should. But keep your business separate from her business. There are legal ramifications if you do not, at least until you are married.
Personal opinion regarding domain names.... It's pointless to "wait for one to expire" 95% of the time. That is unless you pay some domain-sniping company to snatch it up at the first opportunity. The cost for that can be as much as trying to purchase the domain from the existing owner. And many, many, many domain owners and squatters only renew yearly so you THINK it's going to expire soon. But realize they have that 30 day grace period and first rights to reinstate it. You can only "snipe" a domain from someone who is not paying attention to it and ignoring all the renewal emails, for at least 30 days. (Heck, Network Solutions will even CALL an owner when a domain is ready to expire.)
I generally think it is better to find a Top Level Domain that is available (I know it's impossible at times). This can, and often will, have general branding implications. It's great to come up with a "cool name" for your company, but if that TLD is never going to be available because someone else also thought of it, it's still rather pointless.
I, myself, only purchase .com domains for primary web presence. Anything else looks like you couldn't get the .com and clients go looking at the .com. I will use .net or other non-TLD domains for things like client access, or general SEO supporting the .com. But I just don't bother with primary web sites unless they are a .com.