Почитайте про разницу между mouseleave и mouseout
<head>
<style>
div {
display:none;
width:100px;
height:100px;
border: solid 1px blue;
}
</style>
</head>
<body>
<ul>
<li onmouseover="document.getElementById('bi1').style.display='block'"
onmouseleave="document.getElementById('bi1').style.display='none';">
<a id="i1"href="" >11111</a>
</li>
<li onmouseover="document.getElementById('bi2').style.display='block'"
onmouseleave="document.getElementById('bi2').style.display='none';">
<a id="i3"href="" >22222</a>
</li>
<li onmouseover="document.getElementById('bi3').style.display='block'"
onmouseleave="document.getElementById('bi3').style.display='none';">
<a id="i3"href="" >33333</a>
</li>
</ul>
<div id="bi1">1111</div>
<div id="bi2">2222</div>
<div id="bi3">3333</div>
</body>