Assuming your 'Meyer' is one of the so-called "Improved" Meyers, which means it was free of citrus tristeza virus, you may be ok. The older 'Meyer' trees all were symptomless carriers of a particularly nasty strain of CTV, and 'Key' lime is very susceptible as a scion.

So if your tree is CTV-free, the combination could do well. The lime will always be less vigorous, so you may need to do some judicious pruning to keep the 'Meyer' from taking it over and crowding it out. And if the tree ever picks up a CTV infection (It's spread by aphids), you will lose the lime scion, but the 'Meyer' will be unaffected.

To the above excellent advice, I would add the following:

1. Any Mexican/Key lime grafts will be *significantly* less hardy than the "Meyer" lemon stock. So any grafts would probably do better if they were protected by "Meyer" lemon growth above -- such growth would tend to act as a protective umbrella during freezes.

However, as mentioned above, the grafts would probably be less vigorous than the "Meyer" branches and shoots. So the best practice would be to graft the lime onto apically dominant stock shoots.

The problem is, of course, that the most apically dominant portions of your existing tree will be found at the top of the plant!

So you need to find a balance. You might, for example, find a vigorous, watersprout-type shoot, and then trace it down to its base, cut it off low, and put a graft on there. (Or, if you prefer, t-bud some lime down there, and then cut the shoot off.)

2. Meyer lemons are quite ornamental. Mexican/Key limes are much less attractive. They're rather twiggy/scrubby in appearance. Even if you like that look, any lime grafts would be quite difference in appearance than the stock plant growth -- so they would probably stand out, and might mar the ornamental value of your tree.

The ideal way to get around this is to put the lime grafts on whichever side of the "Meyer" that is least visible from your house or yard.