
When Harry looks into the Mirror of Erised, he sees his family. This is all we know.
I think the Mirror of Erised is EXTREMELY exact (after all, only one who wanted to find the stone but not use it would see it, anyone else would only see exactly what they wanted from it)... this is kinda hard to follow.
Harry believes all his relatives are dead, except the Dursleys.
Harry obviously does not desire to see the Dursleys in the mirror or anywhere else.
Harry only sees his dead relatives (thinking that's all his relatives) in the Mirror of Erised.
But there is some really hard-to-get-rid-of evidence that there is more family on Lily's side of the family. Most people discredit this, saying that he wouldn't have HAD to go to the Dursleys if there was anyone else in her family still alive. Not to mention, Harry saw a lot of dead relatives in the mirror. But Harry wouldn't have seen live relatives, since he doesn't believe they exist. His greatest desire would be to see his family who has died, not his living relatives, which he thinks are only the Dursleys.
Also, I believe that the charm Dumbledore and Lily placed on Harry works best with someone with the exact blood line. The only person who would have the exact blood line would be a completely biological brother or sister. Not a half-sibling, cousin, or even grandparent.
Reading about Mark Evans in OotP is what really got me thinking about this, and it's only gotton worse by wondering if Harry has living grandparents. I don't think it's a coincidence that Mark shares the same last name as Lily's before she married.
So if the Mirror of Erised showed his EXACT desire (to see his dead relatives, not his living ones) and the charm relied on an EXACT blood line, it's quite plausible that Lily has other relatives who are not full-blooded siblings, and that one of them could be as close as a half-brother on her father's side (Mark Evans's dad)...