dnd 5e – How do Resurrection spells work with someone who has died of old age?

It had already been mentioned that neither resurrection nor true resurrection can bring someone back to life who has died of old age as it states this clearly in the text of these spells. So, if these spells are cast, they simply fail (PHB, pp. 272,284).

There are other options. If the death from old age has occurred within the past 10 days, then you can use reincarnate (PHB, p.271). This spell does not specify not being able to use reincarnation to bring someone back who died of old age. There is a d100 table which determines your new form, or the DM can simply choose. It says you come back as an adult, so as a DM I would use the age ranges shown in the PHB to guide me.

You touch a dead humanoid or a piece of a dead humanoid. Provided that
the creature has been dead no longer than 10 days, the spell forms a
new adult body for it and then calls the soul to enter that body. If
the target’s soul isn’t free or willing to do so, the spell fails.

This question goes into more detail regarding reincarnate and raise dead:
Can raise dead or reincarnate revive creatures who died of old age?

You could try a wish spell. However an interesting thing in 5e about a wish spell is that it allows to replicate any spell which is 8th level or lower. The text does provide the option of achieving something more difficult, e.g. resurrection of a character that dies from old age, but there may be unintended consequences. Also, there will be a 33% chance the caster will never again be able to cast wish again. This is a serious risk and for that reason I would normally try to accommodate an extraordinary wish as long as it does not ruin or break a campaign. The PC is only likely to be able to make an extraordinary wish 1-3 times.

“You might be able to achieve something beyond the scope of the above
examples. State your wish to the DM as precisely as possible. The DM
has great latitude in ruling what occurs in such an instance; the
greater the wish, the greater the likelihood that something goes
wrong. This spell might simply fail, the effect you desire might only
be partly achieved, or you might suffer some unforeseen consequence as
a result of how you worded the wish.” (PHB, p.288)

Also, if the PC has not yet died of old age but is soon to, you can think of ways of preventing the death in the first place. This questions has some very good answers in it covering this topic:
What magic extends life or grants immortality?

In short, plan ahead to not die of old age. Or, as is my preference use reincarnate as it can be fun to suddenly change races during a campaign. My favourite unexpected reincarnation so far has been a Satyr!