It is plausible if you were looking at just the family but putting them in a society where there are billions of people (assuming the planet is similar to earth) then there should be many immortals depending on when the gene came into existence.
To make it more possible for Lydia to be immortal but her family is not you need to make the possibility of activating the immortality gene as low as possible so that only one in eight billion people expresses the gene (assuming she is the only immortal). To get the probability as low as possible use a combination of the recessive genes and epistasis.
If you have two alleles then for Lydia to have the gene her parents must be carriers and using a traditional punnet square Lydia has a 25% chance of being immortal, her parents won't be immortal and her brother has a large 75% chance of not being immortal.
To make it more plausible for a whole society you should use three alleles instead of two for the life expectancy of a person to explain the variable age of people (you can have even more if you want more variety) and it reduces the number of people carrying the recessive immortality gene.
Having alleles A, B and C. A is dominant to all, medium/normal lifespans. B is recessive to A but dominant to C, it provides long lifespans (think 115 years old). C is recessive to all and provides immortality. AA, AB and AC are all short lives. BB and BC are long lives. CC is immortality.
The parents need to be either AC or BC to make Lydia immortal. Most likely you want both parents being BC as you said they don't have a history of low life expectancy. The distant relatives can be AA, AB or AC since they have low life expectancy.
To lower the possibility of others expressing the immortality gene even if they have the recessive gene then set up some obscure and/or specific conditions. These conditions have happened to Lydia or her parents either before her birth or during her gestation in her mothers womb.
The body is in constant communication with the outside world, taking in information and altering the activation of genes, making a genetic equivalent to a biography of all the things that have happened to you, every school yard fight, every hungry night, broken bone, sunburn, if you are stressed, if there is plenty of food or getting enough sleep, it is all recorded. These changes usually don't occur in the genome of every cell in the body but in the gametes (sperm and egg cells). These changes to gene activation are then carried over to the offspring born from those cells, this gives the offspring a better chance in the environment the parents have experienced by exhibiting genes that will be useful from the genes they have.
An example is then Ireland had a potato blite and a resulting famine, the next few generations had noticeably higher rates of obesity (not horrendously but noticeable), that was because the parents genes thought they would need to store food better to survive.
This same type of situation can be used, the parents could have each had a situation growing up that stressed their bodies to make the DNA think the children may need to live longer, maybe the mother had a tough life growing up with little food or lived on the street, the father could have had a terminal illness while young or even recently before Lydia was born, but recovered (Lydia could have had the same inspiring her fear of dying young)
These events will affect the activation of the genes (anything could affect the gene activation but some events could be known to Lydia and she could tell the doctor and he would know what it meant)
The events don't have to be bad or many, but it is hard to identify good ones when bad events are more noticeable. It is also hard to tie eating breakfast everyday to expressing the immortality gene and it is more what her parents did and do than what she did, maybe they had a type of diet that they preferred of high protein liked a type of fruit with lots of vitamin C in it and so ate a lot of it, the events are very variable for epistasis as it is very hard to say certain events caused certain things when each event has resulted (or not at all) in a different change depending on the person.
You should think about not including the high fertility though as there would be hundreds, if not thousands of immortals if they all were very fertile. For this you can have a separate set of genes and follow the above steps exactly and have it that Lydia hit the genetic lottery.
Also consider the type and extent of this immortality. Is Lydia ageless but heals like normal people or is she unkillable and no wound can stop her. This will tell you how long a person can breed for and pass the gene on and how common it is (again you don't want it to be common).