Will a geforce gtx 1050 work on a lenovo thinkcentre m81?

There's an issue I see and it could be the power supply. The Thinkcenter M81 came with either a 320w or 280w power supply. The 320w should be fine and you should be able to successfully use the GTX 1050 without issue. The 280w power supply is troublesome because it may not cover the wattage needed for the card.

Long story short, you might need a new power supply in order to run the GTX 1050. Even the GTX 1050 recommends a 350w power supply but sometimes that exaggerated.

If you do add a new graphics card, watch the CPU usage. If it's pegged too high then you will need to upgrade the CPU. The Core i5-2400s is weak thanks to it's low clock speed.

Probably. Next question -- is it too slow for your cpu? As long as you're not trying to multi-box a lot of gaems/toons at once, the turbo boost in the i5 should fit with the 1050 ok

Apparently that M81 model has been made for a number of years. Some M81 spec's on Lenovo's website showed PCIe v1.0, but a PDF spec sheet for M81 showed PCIe v2.0. So without knowing what specific model you have I'm guessing that if yours came with WinXP it might be the old standard and upgrading graphic might be wasted. But if it came with Win7 it hopefully has the newer PCIe v2.0 which should work fine. In any case your PCI Express bus should be able to supply the 75 watts that a GTX 1050 (or Ti) requires without external power. But check if you have a suitable aux power connector if a card you are considering has separate power connection.

New cards are PCIe v3.0 which are backwards compatible with v2.0. But when I tried my GTX 750 Ti in an old computer with PCIe v1.0A it seemed to freeze and lag just trying to play flash videos. So maybe PCIe v3.0 is too fast for v1.0, but should work fine with v2.0. I'm currently running a GTX 1060 in my main PC with v2.0 slot, but it requires 6-pin aux power.

It should, but be aware that the i5-2400s is just about the weakest desktop quad core i5 and there are better choices of pcs even if slightly more expensive.
The exact model with Machine type would confirm more about the pc
https://support.lenovo.com/...s/pd013522

US$284 is an excellent choice
https://www.newegg.com/...6883797477
Lenovo Desktop Computer ThinkCentre M82 Intel Core i5 3rd Gen 3470 (3.20 GHz) 8 GB DDR3 1 TB HDD Intel HD Graphics 2500 Windows 10 Pro
https://www.cpubenchmark.net/...u_list.php
i5-2400 @ 3.10GHz5874
i5-2400S @ 2.50GHz4896
i5-2500 @ 3.30GHz6250
i5-3470 @ 3.20GHz6597 <<<
i5-3570 @ 3.40GHz7013
i5-4570 @ 3.20GHz7037
i5-4670 @ 3.40GHz7359
i5-6400 @ 2.70GHz6681
i5-6500 @ 3.20GHz7180
i5-7500 @ 3.40GHz8220

So did you get your computer working with the gTX 1050