The first thing you should decide is how to ask the question. Should you call, send a letter or e-mail, or try to discuss the matter face-to-face? If you're on the phone with your mother all the time, then maybe a call is the best mode for broaching the subject with her. On the other hand, bringing it up when you're having a heart-to-heart over tea is appealing because you can more easily gauge her reaction and respond accordingly.
One advantage to posing the question in a letter or e-mail is that there will be a record of the request and (hopefully) a corresponding record of the response. Those records might come in handy later on if there's any confusion about the will.
Each mode has its pros and cons, and whichever you choose also depends on how you and your relative communicate best. Be sure to consider the matter carefully before you actually ask the question.
The next thing to think about is timing. Some people remain sharp as a bell right up until their final breaths, but many don't. Alzheimer's disease and dementia claim the minds of far too many elderly relatives, and for this reason -- if for no other -- the right time to ask for the heirloom you hope to inherit is as soon as possible. It's better to risk being indelicate early on than to muster up the courage when it's too late. In fact, the earlier you pose the question, the more easily the conversation may go. Since the inevitable sad day will be that much further in the future, the conversation may be less emotional.
Above all, the most important element to consider is your approach to asking the difficult question. Make sure to clarify why the object you're asking for is so important to you. Call upon your memories to create a narrative that your relative can easily understand. For instance, tell your mother how you recall watching her write letters and cards at her desk and how you couldn't wait to be big enough to sit there, too. From there, tell her how you look forward to carrying on the tradition of using the desk for letter-writing and then passing it on down to your own children. Share with her your fears of what might happen to the desk if it were to go to someone else for whom it doesn't hold the same meaning. If it's a valuable antique, explain that you're worried it will be sold and lost to the family forever.
Once you've made sure that your relative understands how much the heirloom means to you, it's up to him or her to decide what to do. The important thing is that you won't regret never having made your request.