DISCLAIMER:Any theory that is posted is just in fun. I am posting these as a reader myself, imagining I'm in your shoes and what you would think. None of these theories have potential spoilers in them and can be taken how you want to take them. They may or may not be true, but that's up to you to decide.
First we must assess where each Hightower is. Lord Damon Hightower and his family are at Sunworth, meanwhile their cousin Ser Daniel Hightower is with Rhaenerys Blackfyre. Don't you find that suspicious? Well let's first see what's going on with Damon Hightower.
One can believe that the Hightowers are backing Rhaenerys Blackfyre and are attempting to bring her to Cephri, maybe to replace Aeyron Archer. How do we know this? Firstly, Ser Daniel Hightower is there with her helping her make her trip back to Cephri. But if we assume he's acting on his own will, we have his letter to Arch Maester Orlon - ' It won’t be long until we are in Cephri, I’m pushing her to it...Anyway, pass this on to my cousin, Damon'. The contents of the letter indicate that Lord Hightower is fully aware of Daniel's involvement with Rhae and that he's encouraging her to come to Cephri. Hightower is complicit in Rhae's plot. We also know that Damon has had conversations with people, such as Edmund in book one, to bring Rhae over to Cephri, 'Hightower was adamant in bringing in the ‘Dragon Queen’ to Cephri, suggesting that a marriage between Aeyron and her would help all parties'. We also have the case of Aeryxs Blackfyre's supposed appearance in the Institute; Maester Colon tells Samuel about it yet the other Maesters deny such existence. Lord Hightower would be made aware of such an appearance, but when Samuel asks him about Aeryxs Damon doesn't know about his presence. Although the text shows that Damon is quick to interrupt Samuel and he also 'hid'. Damon Hightower definitely knows about Aeryxs's appearance.
But why? Why would Damon Hightower do this? What has Aeyron done to him? Well, we must understand the events prior to book one - The Youth Rebellion. Lord Damon's sister, Queen Syrell, was killed by Hoare Lorch and The Mountain on the orders of Tywon Casterly and unbeknownst by Ulthor Young. Aeyron did nothing against them, he let them all live although he exiled them from King's Landing for a time. How could you live knowing that your sister's murders have been allowed to live free and still go ahead and commit murders, Hoare and The Mountain still continue their brutal campaigns of killing. Damon Hightower may not be confident in Aeyron anymore, he can't trust him to do anything. Damon also asks Samuel to 'deliver justice to [him], my sister Queen Syrell, her killers are still around', when Samuel refuses Damon says he'll look to other people then. He clearly doesn't see himself getting justice from Aeyron, so it makes sense that he would turn to Rhaenerys and give her his support, because he views Aeyron as complicit in Syrell's murder.
There may be another factor involved, in which Lord Hightower may have felt undermined by Aeyron. After Nestor Nik's death, Hunter's Hill was vacated - someone needed to move in. Aeyron had Lord Paxter Redwyne take over Hunter's Hill, coming all the way from the Arbour island. You could imagine that you're going to be getting a massive promotion (potentially) but in fact it is another colleague that gets that, and maybe you think that colleague doesn't deserve it and that you do. Lord Hightower may feel like that, he could have two castles to his name but instead Aeyron gives it to another lord, probably another lord that doesn't bring the same amount of money into the economy or hold a big political presence. So it's possible that Lord Hightower may have joined Rhaenerys due to the fact that after the war he wasn't rewarded with Hunter's Hill.
So the big idea that brings Damon to Rhaenerys's side is that she will bring him justice when Aeyron isn't. Aeyron even sends Damon The Mountain's skull, although we find out he's kept The Mountain's body for himself - and Lord Ahriman is seemingly repairing it using alchemy.