King Lear is still relevant today because Shakespeare has a real knack for writing things that contain universal themes. King Lear is no different. One of the core pieces of this play is the struggle over power. Lear gives up his throne, but he clearly still wants to retain some power, as evidenced by his need to hang on to his knights and his expectations that everybody still caters to his every need. The power struggle extends beyond Lear to his children as they try to position themselves to have just a little more than everybody else.
The power discussion of this play also expands into discussing what happens in a power vacuum. Lear's rule had things under control, and his dividing up his kingdom sent his family into strife and disorder; however, the entire kingdom also suffered under the lack of having solid and unified leadership. A reader could easily see parallels between that and anytime power structures switch from one system of government to another. It wouldn't even have to be government related. Private companies can go through turmoil after the head of the company resigns, retires, or dies.
A completely different angle for how the play relates to today is in how the play presents women. This play is heavy on the misogyny. Women are just not portrayed positively throughout most of the play, and it has to be noted that the kingdom was doing relatively fine under male leadership and descended into chaos under female leadership. I do not agree with this thinking or this representation of women; however, I am fully aware how male-centered society is.
A person only has to look at the proportion of women in political leadership versus men. Most films that win best picture awards are films about men, and most video games are focused on male characters. Standard female archetypes that are afforded any semblance of power are either hyper-sexualized femme fatales or overbearing and shrewish harridans. Goneril and Regan are not far off from these tropes, and it is sad to reflect that the stereotypes that were applied to women in power in Shakespeare's time still persist today.
The same basic facts about human nature that motivated people in Shakespeare's time are still true and relevant today. People today, just as Goneril and Regan did, will tell any lie to get the power and wealth they crave. The two daughters swear unremitting love and devotion to their father, only to turn on him cruelly when they get his land and power. This explains the importance, then as now, of not relying on a "handshake" or verbal promises to seal a deal. Once power is transferred, it is too easy for the powerful to forget their promises and do what they want. A legal contract with all obligations specified would have given Lear much more leverage with his daughters, just as it would for a person in a similar situation today.
Another truth which shows the play's relevance is the way power can blind powerful people. Lear has been catered to and treated with respect for so long that he can't imagine life being any other way. He loses sight of the fact that people cater to him because he is king (because of his position) and comes to embrace the mistaken idea that they love him for himself. Today, many powerful people and celebrities also make the mistake of putting too much credence in their "press," which says they are perfect, and forget that they are merely human beings whom other people may not necessarily adore for themselves. The play warns us not to confuse our positional power with our personal worth.
Your question brought to mind the newspapers of a few weeks ago in which Kim Yong Il of North Korea announced the succession of his son, Kim Young-Nam as the new head of state to take over for him. In the same article, Kim Yong Il's brother, Kim Yong Un, was promoted by Kim Yong Il to the rank of four star general. This is a modern case of a madman who, like Lear, has an elevated opinion of himself and chooses his successor based on nepotism. Who knows, perhaps Kim Yong Il had a contest to see which of his sons loved him more -- only kidding.
To answer your question, as long as there are prideful men and women who put themselves before the good of their country and their families, the play KING LEAR will be relevant. Do you remember how they found Saddam Hussein? The former madman ruler of Iraq, who also had his family installed in all of the powerful positions in the country, was reduced to a crazed lunatic-looking man hiding in a hole in the ground. Kind of like the blind King Lear, wandering around lost.
We still have sibling rivalry today, so that issue in the play is also relevant, and we still have heirs fighting over who should get what of a parent's estate. If you read the analysis of the play here on eNotes, you will also find an excellent viewpoint that posits that this play has a very modern, nihilistic view of mankind that is relevant in today's world.