Hi nickntime - Yes, there are several ways of accomplishing this.

One is to build it on a sub-domain or sub-directory. Another is to build on a MAMP/WAMP server. Then once it's built, turn off pretty urls and make a backup of the database and file directory. Now back on the current site; create a new database and user with the same name and credentials, upload the DB backup and upload file directory backup.

Thanks for the quick reply. I will need to build the site on line so the client can view changes as we develop the new site so will need to do the sub folder option (?). Since the clients domain will stay active on the old site until we are ready, I will need to have a URL for the BH server
for me to edit. I get a free domain when I buy my clients hosting. Could (should) I use that as the sub domain?

Is the new site going to be uploaded to the current domain name? If so, than yes, the free domain could be used. If not, I would build on sub-domain of free domain.

I'm confused to why they need new hosting plan when the site is already hosted. You could add a folder to the current directory and point a sub-domain to it. Example: newsite.currentsite.com

I need to be at BlueHost, I've got all my C5 sites located there. Not impressed with the
hosting company they are now using. So what is the best procedure for getting the new site
up and then when ready, connected with their original domain.

Ok. I'm not sure if it's the best way, but I would spend the $15 to transfer the domain to Bluehost (or just adjust the DNS). Build the new site on a subdomain off the BlueHost main domain. Once that is approved and you want to go live, turn off pretty urls and make backups of database and file directory. Create new empty database & user, import subdomain database, upload files to directory. Then edit the database.php file with new credentials.

Their domain is at GoDaddy. I will get the login from them.
If I change the DNS won't that disconnect their current website?
Thanks for your patience, I'm a bit challenged on understanding
some of the tech details.

No problem. Glad to help. Yes, DNS change when you go live with the new site. Not before. If you decide to transfer, just click the 'leave nameservers' checkbox and you won't loose anything.

<Build the new site on a subdomain off the BlueHost main domain>
Is this the free domain I get from BH? If that is the main domain what then, is the subdomain?

Yes, sort of :)

Think of the subdomain as a branch off the main domain. Here's a simple step by step:
-Log into BH
-Scroll down and click 'File Manager'
-Double click 'public_html'
-Click 'Create New Folder' & name the new folder
-Go back to BH home & click 'Domain Manager'
-Click 'subdomain' in the nav
-Enter the name you would like for subdomain
-Enter the name of the folder you created below that

That folder you made will be where you put your C5 files. Do your database stuff, and go to your subdomain address. Should be good to go. Good luck!