<p>In the course of my career I've noticed that developers working on new functionality are, as a rule, more cheerful than these assigned to troubleshooting and fixing bugs.</p>

<p>Good tips on keeping business support a happy? Organising business support in the way that team's morale isn’t hurt?</p>

I have seen this too. 
There are few approaches that I can think of:
1. Give some thought to selection process.
Not everyone have same career aspirations. While some people are always dying to work for the latest technology and challenging projects, there are others who are content at a stable work environment. Picking the person with right KSA (Knowledge, Skills, Attitude) is always central to good person-job match. In this case, you just need to put more stress on the Attitude part.
2. Choose the right lead
Team leads play crucial role in controlling the motivation and morale of the team. There must be an open and frank channel of communication with team lead and upper management. Ideally a person from the senior management should be made the owner of the team in formal organization chart. Team lead should actively try to foster a team-culture that suites the nature of the team.
3. Rewards
Rewards off course are key tool for managing motivational issues. Just make sure the person receiving the rewards understand that the nature of job was also taken into consideration in decision to reward. If this point is not addressed, it is likely that the person will not take this factor into account when perceiving internal equity.