Wet, muddy, heartbreakingly sad looking lawn :(

Hi, it's been a while since I've been on this site, but need your help again please. My garden has always been prone to holding water, and previous winters we have left it knowing that it would dry out in the spring. But, since last year we adopted Scooby, our lovely Springer Spaniel! Now Scooby is quite happy running around a wet garden but it's resulted in one big , muddy mess, with virtually no grass left at all. I'm itching to get it sorted in Spring, and open to suggestions. It's a very very small garden, so I'm not sure if trying to drain it would work (I'd say about 16 foot by 25 foot) , and friends (nongardeners) have suggested giving up the lawn and going for gravel/decking. Wondering whether to empty my borders all together and make do with hanging baskets and pots.Is there any dogowners out there who have views on gravelled gardens? Or suggestions how to get my lovely lawn back? Thank you in anticipation!