Hoping that I can clear my head via suggestions from those more experienced than me...
I am managing a team of 12, responsible for a specific area in IT Operations.
Some of the work is very complex while a large portion of our operation is not. One of my reports, responsible for an area of complexity, has given me his 2 weeks notice. We are already overwhelmed with volume, and involved with multiple transformations, migrations, etc, etc.
I have someone within the team who wants to be promoted into this position, who would require much training in order to be at the required level of productivity. If he is promoted, his area would then require back-filling, training, etc.
Our current operation is lagging far-behind expectations as is, due to the strict off-shoring of positions and continuous addition of services added to our workload. Also, due to customer service issues as a company, we now have several new PM-types who conduct countless meetings regarding escalations (more chiefs and less Indians). My boss (director) is the type that takes-over, rather than advise. So I am reluctant to get help from that direction.
Sorry for the tangled description. But that's how it is in my world right now. I think any advice is going to help at this point. My brain is simply lacking the answers and motivation required to contain this catastrophe ---

Are you being reluctant to allow development?
Sros,
With regards to individual who wants promotion; Correct me, but it sounds as if you are reluctant to allow then to step up.
There maybe a good reason for this - but I cannot see it from the text you have given.
Couple of questions to ask:
1) Would the individual be good at the job?
2) Is it what the individual really wants? This might seem daft given that they have asked - but sometimes people will put their hand up for it just because of more money, better desk, etc rather than it being something they want to do. If you are having the 121 conversations you should already have a fair idea on the answer to this. If you gut feel is that this is a no then take the individual to one side and walk them through the role and why - it might change you mid or theirs.
3) If you ignore the request, will this individual leave - thus leaving you with two people to replace
4) Does the original role need to be replaced? In the Performance Improvement through effective hiring Podcast, Mark talks about using this as an oppertunity to review the team structure, roles and responsibilities. Also worth considering some level of restructuring if the operation is not meeting expectations (a bit more on that below).
In general though, you should always look to promote from internally if you can. If you can't - then fair enough, but you need to be giving this individual a good reason why - and probably more importantly work with them to develop them to be ready for next time.
Reference operations lagging behind expectations; again, on the face of it, I'd say look at the structures and responsibilities. Why is the operation not meeting expectations? Do you (and team) fully understand the expecations?
It does sound rather like the death spiral of failing to keep up with the workload. You are overworked, you fall behind, your bosses get concerned and asks for more reports, you are even more overworked and fall futher behind, etc, etc ...
For background, I've 12+ IT Management experience so I might be able to provide more practical advise with more detail.
Hope the above helps.
Mark
I would reccomend listening
I would reccomend listening to the "How to handle a massive workload increase" series of podcasts.
In regards to filling the open slot, I agree with the previous poster about looking at a reorginization first.
I would also strongly caution you against just promoting the person quickly to stop the short term pain. In the words of Jeffrey J. Fox, "Hire slow and fire fast". The cost and time lost on a poor new hire/promotion far outweighs the short-term pain in most cases.
If you do promote the person
First of all, I agree with Mark's comments -- lots of good stuff in there I don't need to repeat.
I sympathize with your comment that if you promote someone within your team, that just means you have yet another hole to fill. My first role as a manager came when my boss moved into another role and I took over the team. Obviously I wasn't irreplaceable, but I'd been doing a good enough job to be promoted, so it wasn't a trivial issue to backfill myself. However, I'm sure it was easier to find someone else to fill the IC role than to backfill the manager role. In fact, a couple of years later when I moved into another role, they backfilled my position externally and it was a disaster. Much more difficult to step into the manager role than to step into the IC role. I'm sure there are plenty of exceptions, but generally a manager is harder to replace even if it means two people are training in a new role.
I have no idea if your person is ready to be promoted, and if this was unexpected then perhaps they hadn't been developed yet. One thing that made the situation I mentioned above a disaster is that a lot of the solid performers on the team left for other jobs (promotions) when they didn't get my old job. I take pride in the fact that 3 of the 7 members of my team were promoted, but still cringe when I think about not promoting one of them within the team, which caused much more turmoil than necessary.
If you do promote the person
First of all, I agree with Mark's comments -- lots of good stuff in there I don't need to repeat.
I sympathize with your comment that if you promote someone within your team, that just means you have yet another hole to fill. My first role as a manager came when my boss moved into another role and I took over the team. Obviously I wasn't irreplaceable, but I'd been doing a good enough job to be promoted, so it wasn't a trivial issue to backfill myself. However, I'm sure it was easier to find someone else to fill the IC role than to backfill the manager role. In fact, a couple of years later when I moved into another role, they backfilled my position externally and it was a disaster. Much more difficult to step into the manager role than to step into the IC role. I'm sure there are plenty of exceptions, but generally a manager is harder to replace even if it means two people are training in a new role.
I have no idea if your person is ready to be promoted, and if this was unexpected then perhaps they hadn't been developed yet. One thing that made the situation I mentioned above a disaster is that a lot of the solid performers on the team left for other jobs (promotions) when they didn't get my old job. I take pride in the fact that 3 of the 7 members of my team were promoted, but still cringe when I think about not promoting one of them within the team, which caused much more turmoil than necessary.