No cooking oils at all should be used. While saturated fats are very unhealthy as pointed out in M. Arrowsmith's answer, there is a problem with using unsaturated oils as well. We should first note that there is no need to fry food, you can steam or boil your food, and add whole foods containing fats like avocados, walnuts, and chia seeds. There are no proven nutritional health benefits of frying foods in refined oils over a plant based diet that includes all the fats in the form of the above mentioned whole foods.
Conversely, we know from many observational studies that diets that include cooking oils in any form are worse w.r.t. atherosclerosis compared to plant based diets in which no or very small amounts of cooking oils are used. Such diets do contain some amount of fat, this comes mainly from nuts, seeds and certain oily vegetables like avocados. A very recent study of the indigenous Tsimané people reported:
... the Tsimané, a forager-horticulturalist population of the Bolivian Amazon with few coronary artery disease risk factors, have the lowest reported levels of coronary artery disease of any population recorded to date.
Other such results had been obtained previously in other indigenous or rural populations who stick to a low fat diet containing a lot of fruits and vegetables and who get a lot of physical activity. But in the Tsimané study, research have been able to bring CT scanners to the remote locations where the Tsimané live to directly measure coronary artery calcium scores, while the older studies used more indirect methods. E.g. as reported in the landmark article that made the link between cholesterol and heart disease
In the African population of Uganda coronary heart disease is almost non-existent. This statement is confirmed by adequate necropsy evidence. In the Asian community, on the other hand, coronary heart disease is a major problem.
The article goes on to link the differences to the differences in cholesterol level which is then argued to be due to the differences in the percentage of calories coming from oils, it was 30–45% in the Asian community while 10-15% in the African community. Note that the Tsimané also get 14% of their calories from fats.
Then having established that a low fat diet and a generally healthy lifestyle that includes eating lots of vegetables and getting plenty of exercise yields a low rates of heart disease in indigenous populations, one also needs to consider how other indigenous populations fare who eat a high fat diet. E.g. there exists a popular myth about Eskimos having very low rates of heart disease despite eating a high fat, meat based diet containing hardly any fruits and vegetables. But as pointed out here:
During the 1970s, 2 Danish investigators, Bang and Dyerberg, on being informed that the Greenland Eskimos had a low prevalence of coronary artery disease (CAD) set out to study the diet of this population. Bang and Dyerberg described the “Eskimo diet” as consisting of large amounts of seal and whale blubber (ie, fats of animal origin) and suggested that this diet was a key factor in the alleged low incidence of CAD. This was the beginning of a proliferation of studies that focused on the cardioprotective effects of the “Eskimo diet.” In view of data, which accumulated on this topic during the past 40 years, we conducted a review of published literature to examine whether mortality and morbidity due to CAD are indeed lower in Eskimo/Inuit populations compared with their Caucasian counterparts. Most studies found that the Greenland Eskimos and the Canadian and Alaskan Inuit have CAD as often as the non-Eskimo populations. Notably, Bang and Dyerberg's studies from the 1970s did not investigate the prevalence of CAD in this population; however, their reports are still routinely cited as evidence for the cardioprotective effect of the “Eskimo diet.”
So, the known facts on heart disease prevalence and diets in different populations paints a clear picture: using cooking oils is done at your own peril. But what about using only the healthy unsaturated oils, surely we can evade problems by using, say, olive oil? The main problem with this is that you're going to eat much more oil than the typical 10 to 15% of the natural intake that is seen in indigenous populations that are free of heart disease. This especially in case of a 90 year old grandmother as mentioned in the OP's question whose energy intake and use will be quite low. A few tablespoons of oil will already get you over this limit which makes frying food quite difficult. But why would that be a problem?
The problem is due to the "empty calories effect", the larger the fraction of the oil of the total energy intake, the less room will be left for energy in the form of whole grain carbs such as brown rice, whole grain bread and whole grain pasta. Even if you use the most healthy oils known like olive oil, the more you use of such oils the more you're going to miss out on the health benefits of the whole grain carbs. These health benefits come from not only the vitamins and minerals in whole grain carbs, but also from the fibers.
Fibers are converted in the intestines by gut bacteria to short chained fatty acids (SCFAs) which the body uses for a whole host of things. It has been shown that SCFAs reduce cholesterol levels. Besides whole grains, whole foods that contain fats such as walnuts and chia seeds also contain a lot of useful nutrients including fibers. We have to note here that Total dietary fiber intake should be 25 to 30 grams a day from food, not supplements. Currently, dietary fiber intakes among adults in the United States average about 15 grams a day. Indigenous populations like the Tsimané get more than twice the recommended amounts (I checked this from my own diet which is similar to what the Tsimané eat), suggesting that the recommended amount is way too low. A 20 year-old may then be able to boost fiber intake while still using using significant amounts of cooking oils, but what about a sedentary 90 year old?
In summary, cooking oils have no health benefits, we don't need to use them. Using the healthier unsaturated oils is not a good option either compared to not using any cooking oils, because you're going to get less nutrients as a result. This is going to be more of a problem the older and less physically active you are.