Is 'your mom will be sad' actually a good argument to pose to someone thinking of committing suicide? I often come across people pointing out that the close relatives of someone who is thinking about committing suicide will grieve, as an argument against doing it.

However, I've noticed that it is often these precise close relatives that have a lot to do with someone experiencing stress, anxiety and all the related consequences (in trying to meet their expectations; assuming their fears, etc). So, although accepting such an argument as a valid reson against committing suicide at present, couldn't it actually make it worse for them in trying to cope with the underlying problems in the long run? 

I'm aware that each case is different and I'm just sharing my thoughts as a layman, but I'd like to know what a professional would make out of this?
As with many mental health concerns, the situation for each individual is unique and therefore what we do and how we deal with the concern is also unique. Nothing said in this answer is a hard and fast rule, but are loose guidelines, for more information please check out these helpful links (thanks to @ChrisRogers):

ASIST Suicide Intervention, for Canadian
Suicide Prevention Training, for the UK
Suicide Prevention Programs, for the US


I often come across people pointing out that the close relatives of someone who is thinking about committing suicide will grieve, as an argument against doing it.

This is a true fact. Family and friends of people who have committed suicide do grieve for their lost loved one, and often will blame themselves for their death. Here is an interesting article briefly outlining what family can feel when a loved one dies from suicide.
There are many reasons someone might give if they are thinking of killing themselves, and you are correct in your assumption that something like family pressures are one of them, (Linehan, et al, 1983).

Life Over Limb
You asked:

Although accepting such an argument as a valid reason against committing suicide at present, couldn't it actually make it worse for them in trying to cope with the underlying problems in the long run?

The short answer to this is yes, you could talk someone off the ledge with 'people will miss you' and yes it may make the overall depression worse in the long run.
There are two distinct situations that we can discuss:

A person actively carrying out their suicide plan
A person who has suicidal thoughts but is not currently acting on them.

In the first case, your goal as the person who is witnessing the attempted suicide is to save a life. The most important thing is to make sure that the person attempting to commit suicide is not able to complete it. A good analogy is, in a medical situation we save life over limb. Which means that in the moment, if there is imminent risk of death, you do anything to save the life, even if it means cutting off a limb. We do the same thing with suicide intervention. In this case, the long run doesn't matter, because if you don't save the life, there is no way to heal that persons depression/suicidal thoughts. This is what the ASIST and other suicide interventions teach.
In the second case, the goal is different. In this case the goal is to reduce the cause of the suicidal thoughts, Major Depressive Disorder is often (but not always) a diagnosis that produces suicidal thoughts. It would be the role of a counsellor and the client to determine the root cause of suicidal thoughts (ideations) and determine what the client can do to reduce those thoughts.
You are right that if the root cause is family issues, mentioning those issues as a reason not to commit suicide during imminent threat of suicide could cause more long term issues. But I can not stress enough how important life over limb is, when it comes to imminent suicide.
You said:

couldn't it actually make it worse for them in trying to cope with the underlying problems in the long run?

In conclusion, yes it could make it more difficult for that person to deal with the underlying issues, but if that person is not alive they can't deal with the underlying issues. (Life over Limb).

Linehan, M. M., Goodstein, J. L., Nielsen, S. L., &amp; Chiles, J. A. (1983). Reasons for staying alive when you are thinking of killing yourself: the Reasons for Living Inventory. Journal of consulting and clinical psychology, 51(2), 276. doi: 10.1037/0022-006X.51.2.276