Not the most impressive photo, I know.
I hadn't heard of Leomesezia before this; part of the reason is that it's changed names recently (used to be Howeara, which I had heard of). Leomesezia is the nothogenus1 for crosses of
Rodriguezia, Oncidium, and Leochilus. (EDIT: actually I have no idea what Leomesezia is the nothogenus for. Various websites are pointing me in different directions. My best guess is now Gomesa x Leochilus x Rodriguezia. Check the comments to this post for further taxonomic excitement.)
Leomesezia Lava Burst = Leomesezia Mini-Primi x Rodriguezia lanceolata (Ref.)
Vanda Princess Mikasa = Vanda Royal Sapphire x Vanda coerulea
Vanda Royal Sapphire = Vanda Waianae Blue x Vanda Yip Sum Wah
Vanda Waianae Blue = Vanda Rothschildiana x Vanda Helen Paoa
Vanda Yip Sum Wah = Vanda Pukele x Vanda curvifolia
Vanda Rothschildiana = Vanda coerulea x Vanda sanderiana
Vanda Helen Paoa = Vanda sanderiana x Vanda Emily Notley
Vanda Pukele = Vanda Betsy Sumner x Vanda sanderiana
Vanda Emily Notley = Vanda Memoria T. Iwasaki x Vanda tessellata
Vanda Betsy Sumner = Vanda Faustii x Vanda sanderiana
Vanda Memoria T. Iwasaki = Vanda dearei x Vanda tricolor
Vanda Faustii = Vanda Gilbert Triboulet x Vanda luzonica
Vanda Gilbert Triboulet = Vanda coerulea x Vanda tricolor
Vanda Princess Mikasa = (((Vanda coerulea x Vanda sanderiana) x (Vanda sanderiana x ((Vanda dearei x Vanda tricolor) x Vanda tessellata))) x (((((Vanda coerulea x Vanda tricolor) x Vanda luzonica) x Vanda sanderiana) x Vanda sanderiana) x Vanda curvifolia)) x Vanda coerulea
even if describing where your hybrid came from is very complicated and requires lots of parentheses, the resulting orchid is still a Vanda, because all the individual species that went into making it were Vandas.
However, if you make a cross between plants in multiple genera, like Miltonia flavescens and Brassia verrucosa, you can't call the resulting seedlings Miltonias because half of their genes are from Brassia, and you can't call them Brassias because half of their genes are from Miltonia, so you have to invent a genus for them to belong to or else all the other orchid breeders will laugh at you. These invented genera are called nothogenera, from the Greek nothos (pl. nothoi), meaning bastard. Which seems a little judgmental but whatever. In the Miltonia x Brassia example, the resulting seedlings are collectively designated "Bratonia Aristocrat," where Bratonia is the invented nothogenus.
In most plant families, the names of nothogenera are fairly straightforward combinations of the two original genera, like Gasteria x Aloe = Gasteraloe or Cryptanthus x Billbergia = Cryptbergia, but it looks like after a couple of straightforward early attempts (Brassolaelia = Brassia x Laelia; Neoglossum = Ascoglossum x Neofinetia), orchid breeders started running into cases where the obvious simple names for a new combination was already taken by some other, earlier combination, or where the obvious combination was already a word. (Though that didn't stop anybody from naming the cross of Gastrochilus and Doritis "Gastritis," which I find sort of delightful. Alas, there's a debate about whether Doritis is actually its own genus or just a subset of Phalaenopsis. If the latter group prevails, the orchid breeders of the world will forever be cured of their Gastritis.) So, rather than have a bunch of really similar names, they started honoring people by giving them nothogenus names, hence Alexanderara, Fernandezia, and the like.
I don't know where the name Leomesezia came from, so I don't know if it's honoring a person or not. (Google comes up empty for "Leo Mesez" but insists that I probably mean "Leo Mendez," or possibly "Leo meses." I don't mean either of those, but try convincing Google that you know what you're typing and see how far it gets you.) In any case, that's the logic behind nothogenera.