A team member no longer feels their contribution makes an impact on the overall company. How should I motivate this person?
There are really unlimited possibilities for motivation; I will try to conceptualize few of them
On the spot admiration: Everyone knows how to admire for a good work but it is actually all about timing. Some words like "thanks","great work" or anything else. Let him know instantly what he has just achieved.
Involvement in brainstorming: Involvement in different phases of project where inputs are required from the team member will encourage team member to concentrate more on project. The simplest way is to involve him in a brain storming session, ask his opinion and ways to improve procedure/process. You are not forced to use his opinion but asking them will not going to harm you.
Don't compare: Don't compare with other projects running in the company but show them what they are contributing to the organization. Make them understand that even a single penny counts and little drops will make the ocean.
Motivate by example: Motivate your team members by examples (real) not just motivating stories. Hmmm.. I just read a tweet by twitter that they registered 246 tweets on 24th June 2006 which in now days happened in 1/10th of a second. What if the developers/management people on june,2006 has decided that they are not contributing to the organization?
Be creative: Encourage your team member to be creative and come up with ideas. Ask them to surprise you. Appreciate them on their work, teach them if they are going wrong. Creativity can never be neglected.
Communication: Communication is an important tool. Give them a chance to share their issue whether professional or personal. Be a good listener and a friendly adviser. Help them shaping their professional life. Set challenges and goals for them.
There could be several other ways to motivate team member, nothing works better than Self motivation. Do some sessions/learning for self motivation.
Don't forget to grab the feedback from them in some way or another.
First, validate that this low morale exhibited by a single individual is not systemic throughout your organization. Look at your turnover, your sick days, productivity, etc. If you conclude you have a systemic issue, you can introduce some interventions to help, keeping in mind the current school of thought of critical motivators: autonomy, mastery, and purpose. Break those down into concrete examples and find out how your organization's culture is inhibiting them.
If you conclude that this is limited to just this person, then I suggest there is little you can do. The "I feel my contribution does not make an impact..." is either his self diagnosis of his morale and, therefore, not likely to be accurate, or is what he was willing to tell you, which will certainly not be accurate. Either way, the roots of low morale at the individual level is likely deep inside of him versus anything you can offer at the job level. Even if he blamed his low morale on being passed up on a promotion--something you can control--that is usually the cause of being disappointed but not the cause of low morale.
By introducing some of the interventions indicated here, you may see a blip--and he may report improvement--but the sustainability of that is not likely to be long lasting. That would be more of a Hawthorne effect or observation bias at play.
If you have the luxury of an Employee Assistance Program or perhaps some extended leave, suggest that to him so he can do a bit of self examination. Other than that, keep your expectation of his performance high and protect the rest of your team. Low morale is an airborne virus (not literally).