How can I upgrade my GPU for my early 2011 17" MacBook Pro to run Mojave?
To run Mojave on this machine you might to check out macOS Mojave Patcher Tool for Unsupported Macs. A software solution that allows installation – and offers some serious limitations for an unsupported configuration. It does work, but it's probably not worth it in every case. Although you can't beat the price at 0 Dollars… 

Test it out to see if you can live with the limitations inherent to this solution.

The 17 inch being a major obstacle for this plan: 

In 15 inch MacBook Pros you might source a newer logic board that fits into the case. That is probably what "you heard they offer…"
But 17 inch was never made by Apple after your model. So there never will be anything that was once officially supported by Apple and or Mojave.

So your hackish options for that machine are:


making your own logic board (perhaps not overly realistic)
upgrade just the GPU on your existing logic board


The second option is doable (just one example showing the effort and equipment needed). Albeit quite complicated. But it also requires you to find an independent repair shop or other expert willing to do that, and source a replacement GPU. People willing and actually being capable of this will be rare in any case. This is absolutely forbidden by Apple to do for anyone being an Apple Authorized Service Provider. Forget about asking a genius at the bar about this.

You'd need to find a 'hacker'-type to solder that in. That will probably not be very cheap, not overly reliable, and again completely unsupported. One problem is that graphic switching gets in the way. 

Only exchanging the GPU would still leave you with the Intel 3000HD integrated into the Sandy-Bridge CPU. Which is still commonly used then when graphics switching is enabled. 

So upgrading from a  Terascale Radeon HD 6000 series to a Radeon GCN. For example going from a Radeon HD 6490M to a Radeon HD 7730M. Note how confusing AMD named, and re-branded their chips and cards. And compare how these 'newer' but now equally old chips compare to for example newer integrated Intel iGPUs of the type HD Graphics 620.

It also would sensibly require a CPU swap as well. Going to Ivy Bridge, generation 3. Or advanced tinkering with hackintosh configurations, boot loaders, drivers or patches.

With "support" I not only mean "never bring that again near an ASAP" but hardware and software of such a FrankMac (including EFI, smc, OS…) were never tested in this config. The electro-physical connectivity is one thing, the limited compatibility of options Apple decided to ship and support are another. Laptops are so highly integrated these days… I doubt it would be worth it, neither in terms of money, time, effort or ensuing reliability and lifetime of the machine – except for "we made it do".

The realistic run down is:


Stay below 10.14
Use the patcher
Make the machine stationary and use an eGPU (also not supported and obviously taking away mobility)
Buy a newer machine
burn some money on an adventure that may very well ruin the still functioning hardware you have for an exciting experiment with unclear outcomes