I am about to start the branding process of one of our products. There is an interest among the stakeholders for this product to have a separate identity, and they will pay for it. My question, is, should I give a lot of thought to the process of the logo by itself, or should we ask the designer to create all in one while doing the web-site design?
If you are serious about creating a new brand, your first step is to develop that brand in written form. Then comes visual realisation - at which point your question of sequence in design comes up.
The reason for making this point is to help you with the question of selection. How do you choose between two visual representations of your brand? You are trying to optimise both fit and appeal - 'fit' in the sense of conveying aspects of your articulated brand, 'appeal' in the sense of being effective (which typically means being liked and being memorable).
Ideally you might have budget for this brand development activity, so that you can access specialist assistance. Otherwise, you should reach out for a practical methodology.
One framework that I like to use asks you three questions:
Based on the answers, your job (in this framework) is to come up with a paragraph, reduce it to a sentence, shorten it to a phrase and ideally come up with two(!) words that together convey some sense of that whole. Whether you get to such a concentrated essence, what's important is that what you end up with integrates those three aspects.
Well, if you go through a process like this, you have the beginnings of a creative briefing. And you also have a means of evaluating candidate visuals.
Finally it's time to ask, logo or site first?
If you are offering a web service, my suggestion is that you start with the web site as the point of reference. (Or if it's an app, start with that. And so on.) This makes the exercise far more practical, and helps people who don't think of themselves as creative to engage with the discussion.
This sounds controversial, but really it's just 'design unplugged'. The reason that we tend to use 'stylebook' approaches is a matter of compromise. If a visual identity is going to be used in a wide variety of contexts, it needs to be defined in terms of very generally applicable building blocks (typefaces, tones, visual elements). But if the visual identity is primarily for the web, there's no need for such a compromise.
That's not to say that branding design and web site design and the same thing. A good branding-oriented web design is very like a concept car - it sets a direction, and then engineering and customer engagement get involved in the specifics.
If all this sounds very personal, it is. It's not easily crowdsourced. And it wouldn't work on a shoestring budget. So it won't work for every startup.
If you do have the resources, though, it's not just theoretically 'right,' it's the most stimulating, rewarding and effective way to go.
If you're going with a real designer, then the appropriate order of operations is to start with an identity package (logo, color schemes, etc). Reason being, this process should involve several rounds of back-and-forth to truly capture what you want/need to convey with your brand. It takes a knowledgable visual communication professional (designer)to effectively coordinate this process. Once you have that, then it's time for products (letterhead, website, whatever).
However, if you're doing something like 99designs or using your friend's cousin who's good with photoshop, then it won't matter. Just do it all at once, and pick somethign that you like.
From seeing the obvious interest of the stakeholders about the product getting a separate identity, I think that you should hire a Real Designer as Alex stated above, and yes normally you should worry about the identity of your product first -we all had one of those moments of I know that name before, then you see the logo and remember what is it-. so I think that you should start with the (logo, theme), then after that you can start working on website or anything else.
Just wanted to add that if you're working on your logo or brand strategy, you should make sure you don't inadvertently infringe somebody else's logo or trademark. Much better to do this before you start printing materials with your designs only to find out someone else already is using a similar name or image. In the US, you can check the United States Patent & Trademark Office's website: www.uspto.gov, and in the middle trademarks column, click on "Search Marks."
Do the logo first... presumably its a word or phrase? If so... the word and its meaning were chosen for a reason. If its a practical reason, get a practical looking logo and then after a reasonably practical site design. If the word is a creative or inspiring one, get a creative logo and then develop a creative looking website around it.
If you don't have the word or phrase yet... well think about whether your product or even better the user is using it for fun, serious work or other.. and apply the above thinking.