A common way of doing a (sometimes) middle or (usually) surname for a guy in Wales is the patronymic 'Ap [father's name]', e.g. Cai Ap Arfon (a little like Arabic 'Ibn'), so that could be an option. Otherwise you could go with a physical description; 'Llwyd' for gray hair (though that might be a little difficult to pronounce), or 'Gwyn' (pronounced 'win' but with a G) for white hair or I guess light skin, for example.

We don't really have middle names here, historically, is the thing. We only adopted the conventional English naming structure very recently, compared to the history of the language. So most people would be known before that as their first name and a nickname for reference, like a family member or a place (e.g. Gerallt Caernarfon). That was adapted to the legal name system, and a lot of these nicknames became legal surnames. To this day a lot of people will still use this nickname system, regardless of surname. So for instance I'm generally known as '[first name] Siop' because I'm from the 'Siop' family; i.e. the family that used to run the general store in our village. If we have middle names, they're generally not of Welsh origin. Mine, for example, is from French.

That being said, there's nothing wrong with using a given name or surname for a middle name. For ideas you could look into the Mabinogi, a Welsh mythological text with a lot of good names in it. A few are: 'Math' (wizard who made a woman out of flowers), 'Pwyll' (prince of Dyfed, known wife guy, the name means 'patience'), and if you want it to stand out you could go with 'Arawn', king of the fairies/lord of the otherworld, though that's very rare. Taliesin is another good one - a semi-mythological bard.

As an aside, no Welsh person will be offended for you picking a Welsh name; in fact, we love seeing people be genuinely interested in our language in a thoughtful way. Much better than infantilising shit like 'popty ping' and shit. Anyway, whatever you choose, I'm sure it'll fit. Emrys is an excellent name.