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I was recently looking at a chemical equation calculator that balances equations for you.

I came across a reaction that I am unsure about, it goes like this:

B₂H₆ + LiH + PO₄ = LiB₂PO₄H₇

I was researching the name of the product of this reaction for a while, but couldn't find a name for the chemical. I ask you guys this, if this chemical actually exists, what is the name of it?

There is an important concept to be understood underlying this question, something all new chemistry students eventually learn: all sensible chemical reactions will balance, but not all equations that balance make chemical sense.

Balancing is purely mathematical manipulation (solving a system of linear equations), and proper balancing is necessary but not sufficient to represent real chemistry.
As a simple example, $ce{200 H2 + O2 -> 2 H200O}$ balances just fine, but doesn't actually mean anything chemically. Here are a few more examples of proposed chemical reactions which balance correctly, but don't actually represent real chemistry due to more subtle arguments.

The crux of the problem is that it is very easy to write a program which can do mathematical balancing of chemical reactions, but it is exceptionally difficult to write a program which can tell in general whether the reaction actually happens! Chemists have spent the past several hundred years tabulating enormous amounts of information (e.g. physical chemistry) and making qualitative descriptions of chemical behaviour (e.g. atomic models, Lewis dot structures, etc.) so we can make this problem tractable after "getting a feel for it".

Okay, back to your reaction. In your particular case, things are made worse by the fact that one of your reagents, $ce{PO4}$, already doesn't exist in any reasonable conditions. Because this is such an unusual chemical, it's hard to predict exactly how it would react based on the rules-of-thumb we know, as most have been developed to understand more "normal" chemistry. What would most likely happen is that it would tear electrons out of just about anything, such as the hydride ions in $ce{LiH}$. The products could end up as a mixture of $ce{Li3PO4}$, $ce{LiBH4}$, $ce{H2}$ and possibly left over reagents, among others.

The closest chemically valid entity to $ce{PO4}$ is the phosphate anion $ce{PO4^3-}$. The calculator you used doesn't understand this by itself, so you would have to write "PO4{3-}" instead of "PO4" in the input field. If you had done this, since $ce{LiH}$ and $ce{B2H6}$ are true, neutral compounds, then any combination of all the above must have a negative charge too (conservation of charge). Nevertheless, I would still expect the online calculator to fail. My basic guess is that the most favourable outcome would be the formation of $ce{LiBH4}$, with other less important side-reactions.

I've never stumbled upon a "balancing" website that would be checking the correctness of the reaction. This one doesn't even seem to take oxidation state of any element into account as the charges are not balanced (-3 from $ce{PO4^3-}$ on the left and -1 from $ce{LiB2PO4H7^-}$ on the right).

Also, as it's been already ruled out in other answer, these precursors would unlikely result in any borophosphates. However, there is at least one compound characterized that consists of $ce{H,Li,B,O,P}$ only:

• 
  $ce{Li[B3PO6(OH)3]}$, catena-[monoboro-mono-dihydrogendiboratemonohydrogenphosphate]. Hydrothermal synthesis from $ce{LiOH · 2 H2O}$, $ce{P2O5}$ and $ce{B2O3}$, conc. solution in $ce{HCl}$ at $433~mathrm{K}$ [1].

There is also a few dozens of lithium borophosphates with addenda metals, to name a few:

• 
  $ce{LiCd(H2O)2[BP2O8] · H2O}$, lithium cadmium diaqua catena-[monoborodiphosphate]- monohydrate. Hydrothermal synthesis from $ce{CdCl2 · 2.5 H2O}$, $ce{LiOH}$, $ce{H3BO3}$ and $85%$ $ce{H3PO4}$ in deionized water at $443~mathrm{K}$ [2].


• 
  $ce{Li3V2[BP3O12(OH)][HPO4]}$, trilithium divanadium(III) borophosphate hydrogenphosphate. Hydrothermal synthesis from $ce{H3BO3}$, $ce{VCl3}$, $ce{LiH2PO4}$, $ce{LiCl}$ in deionized water at $553~mathrm{K}$ [3].


• 
  $ce{LiCu2[BP2O8(OH)2]}$, lithium dicopper hihydroxoboro-bis(phosphate(V)). Hydrothermal synthesis from $ce{H3BO3}$, $ce{Cu(OAc)2·H2O}$, $ce{LiH2PO4}$ in deionized water at $473~mathrm{K}$ [4].

As you can see, all methods use less volatile precursors and hydrothermal conditions. Feel free to practice with writing and balancing chemical reactions for these real syntheses.

References

1. Hauf, C.; Kniep, R. Crystal Structure of Lithium Catena-[Monoboro-Mono-Dihydrogendiboratemonohydrogenphosphate), $ce{Li[B3PO6(OH)3]}$. Zeitschrift für Kristallographie - New Crystal Structures 1997, 212 (1), 313–314. https://doi.org/10.1524/ncrs.1997.212.1.313. (Open Access)


2. Ge, M.-H.; Mi, J.-X.; Huangm, Y.-X.; Zhao, J.-T.; Kniep, R. Crystal Structure of Lithium Cadmium Diaqua Catena-[Monoborodiphosphate]- Monohydrate, $ce{LiCd(H2O)2[BP2O8] · H2O}$. Zeitschrift für Kristallographie - New Crystal Structures 2003, 218 (JG), 295–296. https://doi.org/10.1524/ncrs.2003.218.jg.295. (Open Access)


3. Lin, Z.-S.; Hoffmann, S.; Huang, Y.-X.; Prots, Y.; Zhao, J.-T.; Kniep, R. Crystal Structure of Trilithium Divanadium(III) Borophosphate Hydrogenphosphate, $ce{Li3V2[BP3O12(OH)][HPO4]}$. Zeitschrift für Kristallographie - New Crystal Structures 2014, 225 (1), 3–4. https://doi.org/10.1524/ncrs.2010.0002. (Open Access)


4. Yang, M.; Li, X.; Yu, J.; Zhu, J.; Liu, X.; Chen, G.; Yan, Y. $ce{LiCu2[BP2O8(OH)2]}$: A Chiral Open-Framework Copper Borophosphate via Spontaneous Asymmetrical Crystallization. Dalton Trans. 2013, 42 (18), 6298–6301. https://doi.org/10.1039/C3DT50591J.

This ion is most likely not the correct product, like the answers say, especially since it is not. However, IUPAC has methods on how to name compounds like these. Take a look at this page in the IUPAC red book.

However, if you were to just name the compound, ignoring the fact that its charge does not balance, you would get something along the lines of:

Lithium diboron phosphate heptahydride.