If you don`t want to deal with giving this person equity another option you could consider is buying the company out. If he is contacting you about a merger then he is motivated and would possibly consider being bought out. Obviously it`s a bit more complicated than that and there are lot of financial figures that you need to look at but it`s an option.
To give you more detailed advice I think we all would need more info about what kind of offer he proposed and the general subscription numbers of his and your websites.
If they are not a serious competitor of yours you could consider doing nothing at all and eventually they will abandon their product or their customers will naturally migrate to your product anyway. You could launch a direct marketing effort to convert their customers which could be less costly than buying them out right. Because if you buy their website and convert it to yours their is bound to be some customer backlash and some would leave your service after you purchase their account.
I would say a buyout or ignoring their offer and focusing on converting their customers through direct marketing efforts would be the best approach.
If you offer a buyout consider all the risks and use those as leverage to the purchase. You can make a low ball offer and the worst they can do is reject it and possibly renegotiate.