Is it reasonable to purchase a professionally designed logo/brand for a research group?
Doing this as soon as you start your research group seems a bit premature, but for a somewhat established group, having some sort of corporate identity is actually quite important:


For other researchers, it is just easier to remember the "Developer Liberation Front" than that "software engineering research group led by Emerson". And, like it or not, being remembered is core in having impact.
Internally, having a consistent identity to which all group members mentally subscribe to helps build team spirit. It is easy to get lost in your own PhD project and feel isolated. Having a corporate identity is just one small puzzle piece towards tying all individual theses into a larger context.



  [from O.R. Mapper's comment] a list of opportunities where you think you might realistically be using that logo?


We apply our logo and general identity very widely. We use it on all websites, presentations, posters, and reports coming out of our team. We have printed coffee mugs and t-shirts (when a large part of the team goes to a conference, we sometimes all show up in the same group-themed shirt). We maintain a Facebook site, a Wordpress blog, and a Twitter handle.


  how much [should I spend]? Which services are the best? What is a reasonable policy?


In design, there is a particularly wide range of "professional". Is paying an artistically inclined student a small sum "professional"? Is using the aforementioned Fiverr professional? Or do you only count if at least a 4-digit sum is changing hands?

In this range, for research groups just like for anybody else, the right "price" should be "the cheapest option that still works for you". If you can find somebody who does a nice logo for free or close to free, go for it. If you can make these cheap Web services work well enough, go for it. If you can't produce a nice logo in any other way, use the service of a local designer (it's still better than not having a logo at a certain point).

When you get a guilty conscience about spending research money badly, then keep in mind that even a small 4-digit sum (about the max. I would spend, personally) is in the larger context of a research group budget not a very big deal. One word of warning, though - typically, not all budgets can be used for something like a group logo (e.g., you clearly can't take it out of your travel budget). Make sure that you actually have a budget that allows you to pay for something like this before you spend the money.