Understand the primary focus of the company you are considering working with.
Some firms approach web relaunches as primarily a development job, others as primarily a content management system job, others as primarily a content generation job, and others primarily as a strategy job. Even if they bid on different kinds of work, most companies typically have a main focus, and understanding what that primary focus is will help you understand what they will be best at doing. Read their proposal, read their blog, talk to their references — understand what the company does best.
Hire beyond the capabilities of your internal team.
You should be hiring a firm to execute a web project in a way that would not be attainable if you kept the work in-house — or opted to work with freelancers instead of a single firm. This doesn’t have to be true for every area of a web relaunch (design, development, CMS implementation, content, and training), but you should have a very clear sense of what the selected firm will help you accomplish that is beyond your internal capabilities.
Everyone critical to the success of the project should help select the partner firm.
If you work on a web team, everyone on the internal core team should participate in the vendor selection. The internal team should feel good about what the outside team is bringing to the table, or dysfunctional things may begin to happen when the project is underway.
Read, re-read, and re-re-read the contracts of competing firms.
The scope of work proposed is always going to look slightly different between firms. Make sure you understand exactly what each firm in question is offering you. Don’t be afraid to ask for revisions from the firm you are secretly rooting for — unless doing so violates your institution’s policies for managing responses.
Understand how the firm approaches content management system (CMS) selection, implementation, and training — assuming these things are part of the project scope.
Your site will live and die by content, and your CMS is the space your content editors will live in. From the response or from interviews, you should be able to gain clarity on how each potential partner firm will select, implement, and train your staff to use a content management system. If you are in the market for an enterprise-level CMS, I would be skeptical of any web design/development company that only offers one solution. There is no such thing as a perfect CMS and one size does not fit everyone.
Follow the rules, of course!
Most institutions have procedures for selecting a partner firm. These procedures often include requirements for writing the RFP, collecting competitive proposals, evaluating the proposals, and setting a budget. The rules should be followed with two strategic objectives in mind — eliminate candidates that aren’t viable in order to get to a short list for comparison and select the firm that is the best fit.